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Mirrors > Home > MPE Home > Th. List > cbvalv | Structured version Visualization version GIF version |
Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2373. See cbvalvw 2031 for a version requiring fewer axioms, to be preferred when sufficient. (Contributed by NM, 5-Aug-1993.) Remove dependency on ax-10 2137 and shorten proof. (Revised by Wolf Lammen, 11-Sep-2023.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cbvalv.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvalv | ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1910 | . 2 ⊢ Ⅎ𝑦𝜑 | |
2 | nfv 1910 | . 2 ⊢ Ⅎ𝑥𝜓 | |
3 | cbvalv.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | cbval 2399 | 1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∀wal 1533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-11 2153 ax-12 2173 ax-13 2373 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1775 df-nf 1779 |
This theorem is referenced by: cbval2vv 2414 cdeqal1 3780 |
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