Theorem rexeqbid 3327

Description: Equality deduction for restricted existential quantifier. See rexeqbidv 3315 for a version based on fewer axioms. (Contributed by Thierry Arnoux, 8-Mar-2017.)

Hypotheses

Ref	Expression
raleqbid.0	⊢ Ⅎ𝑥𝜑
raleqbid.1	⊢ Ⅎ𝑥𝐴
raleqbid.2	⊢ Ⅎ𝑥𝐵
raleqbid.3	⊢ (𝜑 → 𝐴 = 𝐵)
raleqbid.4	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
rexeqbid	⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒))

Proof of Theorem rexeqbid

Step	Hyp	Ref	Expression
1		raleqbid.3	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
2		raleqbid.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3		raleqbid.2	. . . 4 ⊢ Ⅎ𝑥𝐵
4	2, 3	rexeqf 3324	. . 3 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜓))
5	1, 4	syl 17	. 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜓))
6		raleqbid.0	. . 3 ⊢ Ⅎ𝑥𝜑
7		raleqbid.4	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
8	6, 7	rexbid 3248	. 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒))
9	5, 8	bitrd 279	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541  Ⅎwnf 1784  Ⅎwnfc 2881  ∃wrex 3058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ral 3050  df-rex 3059

This theorem is referenced by:  iuneq12df  4971