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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 1458 | . 2 ⊢ (𝜑 → ∀𝑦𝜑) |
3 | cbval.2 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
4 | 3 | nfri 1458 | . 2 ⊢ (𝜓 → ∀𝑥𝜓) |
5 | cbval.3 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
6 | 2, 4, 5 | cbvalh 1684 | 1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1288 Ⅎwnf 1395 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 |
This theorem depends on definitions: df-bi 116 df-nf 1396 |
This theorem is referenced by: sb8 1785 cbval2 1845 sb8eu 1962 abbi 2202 cleqf 2253 cbvralf 2585 ralab2 2780 cbvralcsf 2991 dfss2f 3017 elintab 3705 cbviota 4998 sb8iota 5000 dffun6f 5041 dffun4f 5044 mptfvex 5401 findcard2 6659 findcard2s 6660 |
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