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Mirrors > Home > NFE Home > Th. List > cbvralv | GIF version |
Description: Change the bound variable of a restricted universal quantifier using implicit substitution. (Contributed by NM, 28-Jan-1997.) |
Ref | Expression |
---|---|
cbvralv.1 | ⊢ (x = y → (φ ↔ ψ)) |
Ref | Expression |
---|---|
cbvralv | ⊢ (∀x ∈ A φ ↔ ∀y ∈ A ψ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1619 | . 2 ⊢ Ⅎyφ | |
2 | nfv 1619 | . 2 ⊢ Ⅎxψ | |
3 | cbvralv.1 | . 2 ⊢ (x = y → (φ ↔ ψ)) | |
4 | 1, 2, 3 | cbvral 2831 | 1 ⊢ (∀x ∈ A φ ↔ ∀y ∈ A ψ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∀wral 2614 |
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 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ral 2619 |
This theorem is referenced by: cbvral2v 2843 cbvral3v 2845 reu7 3031 nndisjeq 4429 evenodddisj 4516 nnadjoin 4520 tfinnn 4534 nnc3n3p1 6278 nchoicelem12 6300 nchoicelem17 6305 |
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