3exbidv - Metamath Proof Explorer

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Theorem 3exbidv 1921

Description: Formula-building rule for three existential quantifiers (deduction form). (Contributed by NM, 1-May-1995.)

**Hypothesis**
Ref	Expression
3exbidv.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))

**Assertion**
Ref	Expression
3exbidv	⊢ (𝜑 → (∃𝑥∃𝑦∃𝑧𝜓 ↔ ∃𝑥∃𝑦∃𝑧𝜒))

Distinct variable groups: 𝜑,𝑥 𝜑,𝑦 𝜑,𝑧 Allowed substitution hints: 𝜓(𝑥,𝑦,𝑧) 𝜒(𝑥,𝑦,𝑧)

**Proof of Theorem 3exbidv**
Step	Hyp	Ref	Expression
1		3exbidv.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2	1	exbidv 1917	. 2 ⊢ (𝜑 → (∃𝑧𝜓 ↔ ∃𝑧𝜒))
3	2	2exbidv 1920	1 ⊢ (𝜑 → (∃𝑥∃𝑦∃𝑧𝜓 ↔ ∃𝑥∃𝑦∃𝑧𝜒))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∃wex 1774

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

This theorem depends on definitions: df-bi 206 df-ex 1775

This theorem is referenced by: ceqsex6v 3530 euotd 5510 oprabidw 7446 oprabid 7447 0mpo0 7498 eloprabga 7523 eloprabgaOLD 7524 eloprabi 8062 bnj981 34576 fundcmpsurbijinj 46741

	
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