![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 1530 | . 2 ⊢ (𝜑 → ∀𝑦𝜑) |
3 | cbval.2 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
4 | 3 | nfri 1530 | . 2 ⊢ (𝜓 → ∀𝑥𝜓) |
5 | cbval.3 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
6 | 2, 4, 5 | cbvalh 1764 | 1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∀wal 1362 Ⅎwnf 1471 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 |
This theorem depends on definitions: df-bi 117 df-nf 1472 |
This theorem is referenced by: sb8 1867 cbval2 1933 sb8eu 2051 abbi 2303 cleqf 2357 cbvralf 2710 ralab2 2916 cbvralcsf 3134 dfss2f 3161 elintab 3873 cbviota 5204 sb8iota 5206 dffun6f 5251 dffun4f 5254 mptfvex 5625 findcard2 6921 findcard2s 6922 |
Copyright terms: Public domain | W3C validator |