Theorem rexeqf 3353

Description: Equality theorem for restricted existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. See rexeq 3321 for a version based on fewer axioms. (Contributed by NM, 9-Oct-2003.) (Revised by Andrew Salmon, 11-Jul-2011.) (Proof shortened by Wolf Lammen, 9-Mar-2025.)

Hypotheses

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

Assertion

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

Proof of Theorem rexeqf

Step	Hyp	Ref	Expression
1		raleqf.1	. . . 4 ⊢ Ⅎ𝑥𝐴
2		raleqf.2	. . . 4 ⊢ Ⅎ𝑥𝐵
3	1, 2	raleqf 3352	. . 3 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀𝑥 ∈ 𝐵 ¬ 𝜑))
4		ralnex 3071	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜑)
5		ralnex 3071	. . 3 ⊢ (∀𝑥 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐵 𝜑)
6	3, 4, 5	3bitr3g 313	. 2 ⊢ (𝐴 = 𝐵 → (¬ ∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐵 𝜑))
7	6	con4bid 317	1 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1539  Ⅎwnfc 2889  ∀wral 3060  ∃wrex 3069

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 2707

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 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070

This theorem is referenced by:  rexeqbid  3356  reueq1f  3424  zfrep6  7980  iuneq12daf  32570  indexa  37741  rexeqif  45176