Theorem rexeqif 45007

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 3357	. 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 206   = wceq 1537  Ⅎwnfc 2888  ∃wrex 3072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ral 3064  df-rex 3073

This theorem is referenced by:  rexanuz2nf  45343