Theorem rexeqif 45904

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 3346	. 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 209   = wceq 1570  Ⅎwnfc 2910  ∃wrex 3089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090

This theorem is referenced by:  rexanuz2nf  46226