eqimssd - Metamath Proof Explorer

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Theorem eqimssd 4029

Description: Equality implies inclusion, deduction version. (Contributed by SN, 6-Nov-2024.)

**Hypothesis**
Ref	Expression
eqimssd.1	⊢ (𝜑 → 𝐴 = 𝐵)

**Assertion**
Ref	Expression
eqimssd	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

**Proof of Theorem eqimssd**
Step	Hyp	Ref	Expression
1		eqimssd.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		ssid 3995	. 2 ⊢ 𝐵 ⊆ 𝐵
3	1, 2	eqsstrdi 4027	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ⊆ wss 3940

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-9 2108 ax-ext 2696

This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1774 df-cleq 2717 df-ss 3957

This theorem is referenced by: eqimss 4031 fssrescdmd 7130 sraassab 21803 evls1maplmhm 22303 selvvvval 41882

	
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