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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 5133 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 2899 | . 2 ⊢ Ⅎ𝑦𝐵 | |
2 | nfcv 2899 | . 2 ⊢ Ⅎ𝑥𝐶 | |
3 | cbvmptvg.1 | . 2 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
4 | 1, 2, 3 | cbvmptg 5132 | 1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ↦ cmpt 5110 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-13 2372 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-ex 1787 df-nf 1791 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-v 3400 df-un 3848 df-sn 4517 df-pr 4519 df-op 4523 df-opab 5093 df-mpt 5111 |
This theorem is referenced by: (None) |
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