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Mirrors > Home > MPE Home > Th. List > cbvrex | Structured version Visualization version 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 2751 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2751 | . 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 3142 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 Ⅎwnf 1699 ∃wrex 2897 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 |
This theorem is referenced by: cbvrmo 3146 cbvrexv 3148 cbvrexsv 3159 cbviun 4493 isarep1 5891 elabrex 6405 onminex 6899 boxcutc 7837 indexfi 8157 wdom2d 8368 hsmexlem2 9132 fprodle 14566 iundisj 23123 mbfsup 23237 iundisjf 28784 iundisjfi 28942 voliune 29619 volfiniune 29620 bnj1542 30181 cvmcov 30499 poimirlem24 32603 poimirlem26 32605 indexa 32698 rexrabdioph 36376 rexfrabdioph 36377 elabrexg 38229 dffo3f 38359 disjrnmpt2 38370 stoweidlem31 38924 stoweidlem59 38952 rexsb 39817 cbvrex2 39822 |
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