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| Description: Equality theorem for restricted existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. See rexeq 3321 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.) | 
| Ref | Expression | 
|---|---|
| raleqf.1 | ⊢ Ⅎ𝑥𝐴 | 
| raleqf.2 | ⊢ Ⅎ𝑥𝐵 | 
| Ref | Expression | 
|---|---|
| rexeqf | ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | raleqf.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
| 2 | raleqf.2 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
| 3 | 1, 2 | raleqf 3352 | . . 3 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀𝑥 ∈ 𝐵 ¬ 𝜑)) | 
| 4 | ralnex 3071 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜑) | |
| 5 | ralnex 3071 | . . 3 ⊢ (∀𝑥 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐵 𝜑) | |
| 6 | 3, 4, 5 | 3bitr3g 313 | . 2 ⊢ (𝐴 = 𝐵 → (¬ ∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐵 𝜑)) | 
| 7 | 6 | con4bid 317 | 1 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 = wceq 1539 Ⅎwnfc 2889 ∀wral 3060 ∃wrex 3069 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-nf 1783 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 | 
| This theorem is referenced by: rexeqbid 3356 reueq1f 3424 zfrep6 7980 iuneq12daf 32570 indexa 37741 rexeqif 45176 | 
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