Theorem sseq12i 4010

Description: An equality inference for the subclass relationship. (Contributed by NM, 31-May-1999.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Hypotheses

Ref	Expression
sseq1i.1	⊢ 𝐴 = 𝐵
sseq12i.2	⊢ 𝐶 = 𝐷

Assertion

Ref	Expression
sseq12i	⊢ (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷)

Proof of Theorem sseq12i

Step	Hyp	Ref	Expression
1		sseq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		sseq12i.2	. 2 ⊢ 𝐶 = 𝐷
3		sseq12 4007	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷))
4	1, 2, 3	mp2an 691	1 ⊢ (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷)

Syntax hints:   ↔ wb 205   = wceq 1534   ⊆ wss 3947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3473  df-in 3954  df-ss 3964

This theorem is referenced by:  3sstr3i  4022  3sstr4i  4023  3sstr3g  4024  3sstr4g  4025  ss2rab  4066  rabsssn  4671  fldhmsubc  20673  issubgr  29097  pjordi  31996  mdsldmd1i  32154  iuninc  32364  cvmlift2lem12  34924  brtrclfv2  43157  nzss  43754  hoidmvle  45988  fldhmsubcALTV  47395

Otomatik - 173.255.232.114 CloudFlare DNS Türk Telekom DNS Google DNS Open DNS

OSZAR »