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Mirrors > Home > ILE Home > Th. List > cbval | GIF version |
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) |
Ref | Expression |
---|---|
cbval.1 | ⊢ Ⅎ𝑦𝜑 |
cbval.2 | ⊢ Ⅎ𝑥𝜓 |
cbval.3 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbval | ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbval.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | nfri 1452 | . 2 ⊢ (𝜑 → ∀𝑦𝜑) |
3 | cbval.2 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
4 | 3 | nfri 1452 | . 2 ⊢ (𝜓 → ∀𝑥𝜓) |
5 | cbval.3 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
6 | 2, 4, 5 | cbvalh 1676 | 1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∀wal 1282 Ⅎwnf 1389 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 |
This theorem depends on definitions: df-bi 115 df-nf 1390 |
This theorem is referenced by: sb8 1777 cbval2 1837 sb8eu 1954 abbi 2192 cleqf 2242 cbvralf 2571 ralab2 2756 cbvralcsf 2964 dfss2f 2990 elintab 3647 cbviota 4892 sb8iota 4894 dffun6f 4935 dffun4f 4938 mptfvex 5277 findcard2 6373 findcard2s 6374 |
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