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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 1507 | . 2 ⊢ (𝜑 → ∀𝑦𝜑) | |
2 | ax-17 1507 | . 2 ⊢ (𝜓 → ∀𝑥𝜓) | |
3 | cbvalv.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | cbvalh 1727 | 1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1330 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 |
This theorem depends on definitions: df-bi 116 df-nf 1438 |
This theorem is referenced by: cbvalvw 1892 nfcjust 2270 cdeqal1 2904 dfss4st 3314 zfpow 4107 tfisi 4509 acexmid 5781 tfrlem3-2d 6217 tfrlemi1 6237 tfrexlem 6239 tfr1onlemaccex 6253 tfrcllemaccex 6266 findcard 6790 fisseneq 6828 genprndl 7353 genprndu 7354 zfz1iso 10616 |
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