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Theorem cbvmpt2x2 40625
 Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpt2 6389 allows to be a function of , analogous to cbvmpt2x 6388. (Contributed by AV, 30-Mar-2019.)
Hypotheses
Ref Expression
cbvmpt2x2.1
cbvmpt2x2.2
cbvmpt2x2.3
cbvmpt2x2.4
cbvmpt2x2.5
cbvmpt2x2.6
cbvmpt2x2.7
cbvmpt2x2.8
Assertion
Ref Expression
cbvmpt2x2
Distinct variable groups:   ,,,   ,   ,,,,   ,
Allowed substitution hints:   (,,)   (,,,)   (,,)   (,,,)

Proof of Theorem cbvmpt2x2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfv 1769 . . . . 5
2 nfv 1769 . . . . 5
31, 2nfan 2031 . . . 4
4 cbvmpt2x2.4 . . . . 5
54nfeq2 2627 . . . 4
63, 5nfan 2031 . . 3
7 cbvmpt2x2.1 . . . . . 6
87nfcri 2606 . . . . 5
9 nfv 1769 . . . . 5
108, 9nfan 2031 . . . 4
11 cbvmpt2x2.3 . . . . 5
1211nfeq2 2627 . . . 4
1310, 12nfan 2031 . . 3
14 nfv 1769 . . . . 5
15 nfv 1769 . . . . 5
1614, 15nfan 2031 . . . 4
17 cbvmpt2x2.5 . . . . 5
1817nfeq2 2627 . . . 4
1916, 18nfan 2031 . . 3
20 cbvmpt2x2.2 . . . . . 6
2120nfcri 2606 . . . . 5
22 nfv 1769 . . . . 5
2321, 22nfan 2031 . . . 4
24 cbvmpt2x2.6 . . . . 5
2524nfeq2 2627 . . . 4
2623, 25nfan 2031 . . 3
27 eleq1 2537 . . . . . 6
28 cbvmpt2x2.7 . . . . . . 7
2928eleq2d 2534 . . . . . 6
3027, 29sylan9bb 714 . . . . 5
31 simpr 468 . . . . . 6
3231eleq1d 2533 . . . . 5
3330, 32anbi12d 725 . . . 4
34 cbvmpt2x2.8 . . . . . 6
3534ancoms 460 . . . . 5
3635eqeq2d 2481 . . . 4
3733, 36anbi12d 725 . . 3
386, 13, 19, 26, 37cbvoprab12 6384 . 2
39 df-mpt2 6313 . 2
40 df-mpt2 6313 . 2
4138, 39, 403eqtr4i 2503 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 376   wceq 1452   wcel 1904  wnfc 2599  coprab 6309   cmpt2 6310 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-opab 4455  df-oprab 6312  df-mpt2 6313 This theorem is referenced by:  dmmpt2ssx2  40626
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