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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 1506 | . 2 ⊢ (𝜑 → ∀𝑦𝜑) | |
2 | ax-17 1506 | . 2 ⊢ (𝜓 → ∀𝑥𝜓) | |
3 | cbvalv.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | cbvalh 1726 | 1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1329 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 |
This theorem depends on definitions: df-bi 116 df-nf 1437 |
This theorem is referenced by: cbvalvw 1891 nfcjust 2269 cdeqal1 2900 dfss4st 3309 zfpow 4099 tfisi 4501 acexmid 5773 tfrlem3-2d 6209 tfrlemi1 6229 tfrexlem 6231 tfr1onlemaccex 6245 tfrcllemaccex 6258 findcard 6782 fisseneq 6820 genprndl 7329 genprndu 7330 zfz1iso 10584 |
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