Theorem rexeqif 45944

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 3348	. 2 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑))
5	1, 4	ax-mp 5	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570  Ⅎwnfc 2912  ∃wrex 3091

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3082  df-rex 3092

This theorem is used by:  rexanuz2nf  46266