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Mirrors > Home > MPE Home > Th. List > cbvmptg | Structured version Visualization version GIF version |
Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2389. See cbvmpt 5160 for a version with more disjoint variable conditions, but not requiring ax-13 2389. (Contributed by NM, 11-Sep-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cbvmptg.1 | ⊢ Ⅎ𝑦𝐵 |
cbvmptg.2 | ⊢ Ⅎ𝑥𝐶 |
cbvmptg.3 | ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
cbvmptg | ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2976 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2976 | . 2 ⊢ Ⅎ𝑦𝐴 | |
3 | cbvmptg.1 | . 2 ⊢ Ⅎ𝑦𝐵 | |
4 | cbvmptg.2 | . 2 ⊢ Ⅎ𝑥𝐶 | |
5 | cbvmptg.3 | . 2 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
6 | 1, 2, 3, 4, 5 | cbvmptfg 5159 | 1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 Ⅎwnfc 2960 ↦ cmpt 5139 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-13 2389 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-opab 5122 df-mpt 5140 |
This theorem is referenced by: cbvmptvg 5163 |
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