Theorem rexeqf 3343

Description: Equality theorem for restricted existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. See rexeq 3315 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 3342	. . 3 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀𝑥 ∈ 𝐵 ¬ 𝜑))
4		ralnex 3087	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜑)
5		ralnex 3087	. . 3 ⊢ (∀𝑥 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐵 𝜑)
6	3, 4, 5	3bitr3g 315	. 2 ⊢ (𝐴 = 𝐵 → (¬ ∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐵 𝜑))
7	6	con4bid 319	1 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   = wceq 1559  Ⅎwnfc 2908  ∀wral 3075  ∃wrex 3085

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-ex 1799  df-nf 1803  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3076  df-rex 3086

This theorem is referenced by:  rexeqbid  3345  reueq1f  3404  zfrep6OLD  7931  iuneq12daf  32716  indexa  38193  rexeqif  45705