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Mirrors > Home > NFE Home > Th. List > cbvrex | GIF version |
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Ref | Expression |
---|---|
cbvral.1 | ⊢ Ⅎyφ |
cbvral.2 | ⊢ Ⅎxψ |
cbvral.3 | ⊢ (x = y → (φ ↔ ψ)) |
Ref | Expression |
---|---|
cbvrex | ⊢ (∃x ∈ A φ ↔ ∃y ∈ A ψ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2489 | . 2 ⊢ ℲxA | |
2 | nfcv 2489 | . 2 ⊢ ℲyA | |
3 | cbvral.1 | . 2 ⊢ Ⅎyφ | |
4 | cbvral.2 | . 2 ⊢ Ⅎxψ | |
5 | cbvral.3 | . 2 ⊢ (x = y → (φ ↔ ψ)) | |
6 | 1, 2, 3, 4, 5 | cbvrexf 2830 | 1 ⊢ (∃x ∈ A φ ↔ ∃y ∈ A ψ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 Ⅎwnf 1544 ∃wrex 2615 |
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 df-rex 2620 |
This theorem is referenced by: cbvrmo 2834 cbvrexv 2836 cbvrexsv 2847 cbviun 4003 |
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