![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2371. See cbvmpt 5220 for a version with more disjoint variable conditions, but not requiring ax-13 2371. (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 2904 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2904 | . 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 5219 | 1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 Ⅎwnfc 2884 ↦ cmpt 5192 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-13 2371 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-opab 5172 df-mpt 5193 |
This theorem is referenced by: cbvmptvg 5224 |
Copyright terms: Public domain | W3C validator |