Theorem bj-3exbi 31785

Description: Closed form of 3exbii 1766. (Contributed by BJ, 6-May-2019.)

Assertion

Ref	Expression
bj-3exbi	⊢ (∀𝑥∀𝑦∀𝑧(𝜑 ↔ 𝜓) → (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑥∃𝑦∃𝑧𝜓))

Proof of Theorem bj-3exbi

Step	Hyp	Ref	Expression
1		exbi 1762	. . 3 ⊢ (∀𝑧(𝜑 ↔ 𝜓) → (∃𝑧𝜑 ↔ ∃𝑧𝜓))
2	1	2alimi 1731	. 2 ⊢ (∀𝑥∀𝑦∀𝑧(𝜑 ↔ 𝜓) → ∀𝑥∀𝑦(∃𝑧𝜑 ↔ ∃𝑧𝜓))
3		bj-2exbi 31784	. 2 ⊢ (∀𝑥∀𝑦(∃𝑧𝜑 ↔ ∃𝑧𝜓) → (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑥∃𝑦∃𝑧𝜓))
4	2, 3	syl 17	1 ⊢ (∀𝑥∀𝑦∀𝑧(𝜑 ↔ 𝜓) → (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑥∃𝑦∃𝑧𝜓))

Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728

This theorem depends on definitions:  df-bi 196  df-ex 1696

This theorem is referenced by: (None)