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Mirrors > Home > MPE Home > Th. List > cbvmptvg | Structured version Visualization version GIF version |
Description: Rule to change the bound variable in a maps-to function, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2372. See cbvmptv 5187 for a version with more disjoint variable conditions, but not requiring ax-13 2372. (Contributed by Mario Carneiro, 19-Feb-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cbvmptvg.1 | ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
cbvmptvg | ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2907 | . 2 ⊢ Ⅎ𝑦𝐵 | |
2 | nfcv 2907 | . 2 ⊢ Ⅎ𝑥𝐶 | |
3 | cbvmptvg.1 | . 2 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
4 | 1, 2, 3 | cbvmptg 5186 | 1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ↦ cmpt 5157 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-13 2372 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-opab 5137 df-mpt 5158 |
This theorem is referenced by: (None) |
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