Theorem rexeqf 3353

Description: Equality theorem for restricted existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. See rexeq 3325 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 3097	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜑)
5		ralnex 3097	. . 3 ⊢ (∀𝑥 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐵 𝜑)
6	3, 4, 5	3bitr3g 316	. 2 ⊢ (𝐴 = 𝐵 → (¬ ∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐵 𝜑))
7	6	con4bid 320	1 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1567  Ⅎwnfc 2916  ∀wral 3085  ∃wrex 3095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-nf 1811  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-rex 3096

This theorem is referenced by:  rexeqbid  3355  reueq1f  3414  zfrep6OLD  7948  iuneq12daf  32838  indexa  38267  rexeqif  45771