Theorem rexeqif 45613

Description: Equality inference for restricted existential quantifier. (Contributed by Glauco Siliprandi, 15-Feb-2025.)

Hypotheses

Ref	Expression
rexeqif.1	⊢ Ⅎ𝑥𝐴
rexeqif.2	⊢ Ⅎ𝑥𝐵
rexeqif.3	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
rexeqif	⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑)

Proof of Theorem rexeqif

Step	Hyp	Ref	Expression
1		rexeqif.3	. 2 ⊢ 𝐴 = 𝐵
2		rexeqif.1	. . 3 ⊢ Ⅎ𝑥𝐴
3		rexeqif.2	. . 3 ⊢ Ⅎ𝑥𝐵
4	2, 3	rexeqf 3321	. 2 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑))
5	1, 4	ax-mp 5	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑)

Syntax hints:   ↔ wb 207   = wceq 1547  Ⅎwnfc 2886  ∃wrex 3063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-nf 1791  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ral 3054  df-rex 3064

This theorem is referenced by:  rexanuz2nf  45935