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Mirrors > Home > MPE Home > Th. List > cbvalv1 | Structured version Visualization version GIF version |
Description: Rule used to change bound variables, using implicit substitution. Version of cbval 2398 with a disjoint variable condition, which does not require ax-13 2372. See cbvalvw 2040 for a version with two more disjoint variable conditions, requiring fewer axioms, and cbvalv 2400 for another variant. (Contributed by NM, 13-May-1993.) (Revised by BJ, 31-May-2019.) |
Ref | Expression |
---|---|
cbvalv1.nf1 | ⊢ Ⅎ𝑦𝜑 |
cbvalv1.nf2 | ⊢ Ⅎ𝑥𝜓 |
cbvalv1.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvalv1 | ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvalv1.nf1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
2 | cbvalv1.nf2 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
3 | cbvalv1.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 3 | biimpd 228 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
5 | 1, 2, 4 | cbv3v 2332 | . 2 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) |
6 | 3 | biimprd 247 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜓 → 𝜑)) |
7 | 6 | equcoms 2024 | . . 3 ⊢ (𝑦 = 𝑥 → (𝜓 → 𝜑)) |
8 | 2, 1, 7 | cbv3v 2332 | . 2 ⊢ (∀𝑦𝜓 → ∀𝑥𝜑) |
9 | 5, 8 | impbii 208 | 1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1540 Ⅎwnf 1786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-11 2155 ax-12 2172 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-nf 1787 |
This theorem is referenced by: cbvexv1 2339 cbval2v 2340 sb8fOLD 2351 sbbib 2358 cbvsbvf 2361 sb8eulem 2593 cbvmow 2598 abbib 2805 cleqh 2864 cleqf 2935 cbvralfw 3302 cbvralfwOLD 3321 cbvralf 3357 ralab2 3694 cbvralcsf 3939 dfss2f 3973 ab0OLD 4376 elintabOLD 4964 reusv2lem4 5400 cbviotaw 6503 cbviota 6506 sb8iota 6508 dffun6f 6562 findcard2 9164 findcard2OLD 9284 aceq1 10112 bnj1385 33843 sbcalf 36982 alrimii 36987 aomclem6 41801 rababg 42325 |
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