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Theorem cbvmptfg 5157
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 2384. See cbvmptf 5156 for a version with more disjoint variable conditions, but not requiring ax-13 2384. (Contributed by NM, 11-Sep-2011.) (Revised by Thierry Arnoux, 9-Mar-2017.) (New usage is discouraged.)
Hypotheses
Ref Expression
cbvmptfg.1 𝑥𝐴
cbvmptfg.2 𝑦𝐴
cbvmptfg.3 𝑦𝐵
cbvmptfg.4 𝑥𝐶
cbvmptfg.5 (𝑥 = 𝑦𝐵 = 𝐶)
Assertion
Ref Expression
cbvmptfg (𝑥𝐴𝐵) = (𝑦𝐴𝐶)

Proof of Theorem cbvmptfg
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1909 . . . 4 𝑤(𝑥𝐴𝑧 = 𝐵)
2 cbvmptfg.1 . . . . . 6 𝑥𝐴
32nfcri 2969 . . . . 5 𝑥 𝑤𝐴
4 nfs1v 2267 . . . . 5 𝑥[𝑤 / 𝑥]𝑧 = 𝐵
53, 4nfan 1894 . . . 4 𝑥(𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)
6 eleq1w 2893 . . . . 5 (𝑥 = 𝑤 → (𝑥𝐴𝑤𝐴))
7 sbequ12 2246 . . . . 5 (𝑥 = 𝑤 → (𝑧 = 𝐵 ↔ [𝑤 / 𝑥]𝑧 = 𝐵))
86, 7anbi12d 632 . . . 4 (𝑥 = 𝑤 → ((𝑥𝐴𝑧 = 𝐵) ↔ (𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)))
91, 5, 8cbvopab1g 5131 . . 3 {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧 = 𝐵)} = {⟨𝑤, 𝑧⟩ ∣ (𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)}
10 cbvmptfg.2 . . . . . 6 𝑦𝐴
1110nfcri 2969 . . . . 5 𝑦 𝑤𝐴
12 cbvmptfg.3 . . . . . . 7 𝑦𝐵
1312nfeq2 2993 . . . . . 6 𝑦 𝑧 = 𝐵
1413nfsb 2559 . . . . 5 𝑦[𝑤 / 𝑥]𝑧 = 𝐵
1511, 14nfan 1894 . . . 4 𝑦(𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)
16 nfv 1909 . . . 4 𝑤(𝑦𝐴𝑧 = 𝐶)
17 eleq1w 2893 . . . . 5 (𝑤 = 𝑦 → (𝑤𝐴𝑦𝐴))
18 sbequ 2084 . . . . . 6 (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝑧 = 𝐵 ↔ [𝑦 / 𝑥]𝑧 = 𝐵))
19 cbvmptfg.4 . . . . . . . 8 𝑥𝐶
2019nfeq2 2993 . . . . . . 7 𝑥 𝑧 = 𝐶
21 cbvmptfg.5 . . . . . . . 8 (𝑥 = 𝑦𝐵 = 𝐶)
2221eqeq2d 2830 . . . . . . 7 (𝑥 = 𝑦 → (𝑧 = 𝐵𝑧 = 𝐶))
2320, 22sbie 2538 . . . . . 6 ([𝑦 / 𝑥]𝑧 = 𝐵𝑧 = 𝐶)
2418, 23syl6bb 289 . . . . 5 (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝑧 = 𝐵𝑧 = 𝐶))
2517, 24anbi12d 632 . . . 4 (𝑤 = 𝑦 → ((𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵) ↔ (𝑦𝐴𝑧 = 𝐶)))
2615, 16, 25cbvopab1g 5131 . . 3 {⟨𝑤, 𝑧⟩ ∣ (𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧 = 𝐶)}
279, 26eqtri 2842 . 2 {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧 = 𝐵)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧 = 𝐶)}
28 df-mpt 5138 . 2 (𝑥𝐴𝐵) = {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧 = 𝐵)}
29 df-mpt 5138 . 2 (𝑦𝐴𝐶) = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧 = 𝐶)}
3027, 28, 293eqtr4i 2852 1 (𝑥𝐴𝐵) = (𝑦𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1531  [wsb 2063  wcel 2108  wnfc 2959  {copab 5119  cmpt 5137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-13 2384  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-opab 5120  df-mpt 5138
This theorem is referenced by:  cbvmptg  5159
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