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Mirrors > Home > MPE Home > Th. List > rexeqf | Structured version Visualization version GIF version |
Description: Equality theorem for restricted existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. See rexeq 3320 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 3351 | . . 3 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀𝑥 ∈ 𝐵 ¬ 𝜑)) |
4 | ralnex 3070 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜑) | |
5 | ralnex 3070 | . . 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 1537 Ⅎwnfc 2888 ∀wral 3059 ∃wrex 3068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 |
This theorem is referenced by: rexeqbid 3355 reueq1f 3422 zfrep6 7978 iuneq12daf 32577 indexa 37720 rexeqif 45109 |
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