Theorem rexeqfOLD 3332

Description: Obsolete version of rexeqf 3331 as of 9-Mar-2025. (Contributed by NM, 9-Oct-2003.) (Revised by Andrew Salmon, 11-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
raleqf.1	⊢ Ⅎ𝑥𝐴
raleqf.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
rexeqfOLD	⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑))

Proof of Theorem rexeqfOLD

Step	Hyp	Ref	Expression
1		raleqf.1	. . . 4 ⊢ Ⅎ𝑥𝐴
2		raleqf.2	. . . 4 ⊢ Ⅎ𝑥𝐵
3	1, 2	nfeq 2911	. . 3 ⊢ Ⅎ𝑥 𝐴 = 𝐵
4		eleq2 2822	. . . 4 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
5	4	anbi1d 631	. . 3 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ 𝜑)))
6	3, 5	exbid 2222	. 2 ⊢ (𝐴 = 𝐵 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)))
7		df-rex 3060	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
8		df-rex 3060	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
9	6, 7, 8	3bitr4g 314	1 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539  ∃wex 1778   ∈ wcel 2107  Ⅎwnfc 2882  ∃wrex 3059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-nf 1783  df-cleq 2726  df-clel 2808  df-nfc 2884  df-rex 3060

This theorem is referenced by: (None)