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Mirrors > Home > NFE Home > Th. List > cbval2v | GIF version |
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 4-Feb-2005.) |
Ref | Expression |
---|---|
cbval2v.1 | ⊢ ((x = z ∧ y = w) → (φ ↔ ψ)) |
Ref | Expression |
---|---|
cbval2v | ⊢ (∀x∀yφ ↔ ∀z∀wψ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1619 | . 2 ⊢ Ⅎzφ | |
2 | nfv 1619 | . 2 ⊢ Ⅎwφ | |
3 | nfv 1619 | . 2 ⊢ Ⅎxψ | |
4 | nfv 1619 | . 2 ⊢ Ⅎyψ | |
5 | cbval2v.1 | . 2 ⊢ ((x = z ∧ y = w) → (φ ↔ ψ)) | |
6 | 1, 2, 3, 4, 5 | cbval2 2004 | 1 ⊢ (∀x∀yφ ↔ ∀z∀wψ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 ∀wal 1540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 |
This theorem depends on definitions: df-bi 177 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 |
This theorem is referenced by: nnsucelr 4428 ssfin 4470 fnfrec 6320 |
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