![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > cbvalv | GIF version |
Description: Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
cbvalv.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvalv | ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-17 1419 | . 2 ⊢ (𝜑 → ∀𝑦𝜑) | |
2 | ax-17 1419 | . 2 ⊢ (𝜓 → ∀𝑥𝜓) | |
3 | cbvalv.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | cbvalh 1636 | 1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∀wal 1241 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 |
This theorem depends on definitions: df-bi 110 df-nf 1350 |
This theorem is referenced by: nfcjust 2166 cdeqal1 2755 zfpow 3928 tfisi 4310 acexmid 5511 tfrlem3-2d 5928 tfrlemi1 5946 tfrexlem 5948 findcard 6345 genprndl 6619 genprndu 6620 |
Copyright terms: Public domain | W3C validator |