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Mirrors > Home > ILE Home > Th. List > cbvex | GIF version |
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
cbvex.1 | ⊢ Ⅎ𝑦𝜑 |
cbvex.2 | ⊢ Ⅎ𝑥𝜓 |
cbvex.3 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvex | ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvex.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | nfri 1519 | . 2 ⊢ (𝜑 → ∀𝑦𝜑) |
3 | cbvex.2 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
4 | 3 | nfri 1519 | . 2 ⊢ (𝜓 → ∀𝑥𝜓) |
5 | cbvex.3 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
6 | 2, 4, 5 | cbvexh 1755 | 1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 Ⅎwnf 1460 ∃wex 1492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 |
This theorem depends on definitions: df-bi 117 df-nf 1461 |
This theorem is referenced by: sb8e 1857 cbvex2 1922 cbvmo 2066 mo23 2067 clelab 2303 cbvrexf 2697 issetf 2744 eqvincf 2862 rexab2 2903 cbvrexcsf 3120 abn0m 3448 rabn0m 3450 euabsn 3662 eluniab 3821 cbvopab1 4076 cbvopab2 4077 cbvopab1s 4078 intexabim 4152 iinexgm 4154 opeliunxp 4681 dfdmf 4820 dfrnf 4868 elrnmpt1 4878 cbvoprab1 5946 cbvoprab2 5947 opabex3d 6121 opabex3 6122 seq3f1olemp 10501 fsum2dlemstep 11441 bdsepnfALT 14577 strcollnfALT 14674 |
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