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Theorem cbvmpt2x 5678
 Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpt2 5679 allows to be a function of . (Contributed by NM, 29-Dec-2014.)
Hypotheses
Ref Expression
cbvmpt2x.1
cbvmpt2x.2
cbvmpt2x.3
cbvmpt2x.4
cbvmpt2x.5
cbvmpt2x.6
cbvmpt2x.7
cbvmpt2x.8
Assertion
Ref Expression
cbvmpt2x
Distinct variable groups:   ,,,,   ,   ,
Allowed substitution hints:   (,,)   (,,,)   (,,)   (,,,)

Proof of Theorem cbvmpt2x
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfv 1619 . . . . 5
2 cbvmpt2x.1 . . . . . 6
32nfcri 2483 . . . . 5
41, 3nfan 1824 . . . 4
5 cbvmpt2x.3 . . . . 5
65nfeq2 2500 . . . 4
74, 6nfan 1824 . . 3
8 nfv 1619 . . . . 5
9 nfcv 2489 . . . . . 6
109nfcri 2483 . . . . 5
118, 10nfan 1824 . . . 4
12 cbvmpt2x.4 . . . . 5
1312nfeq2 2500 . . . 4
1411, 13nfan 1824 . . 3
15 nfv 1619 . . . . 5
16 cbvmpt2x.2 . . . . . 6
1716nfcri 2483 . . . . 5
1815, 17nfan 1824 . . . 4
19 cbvmpt2x.5 . . . . 5
2019nfeq2 2500 . . . 4
2118, 20nfan 1824 . . 3
22 nfv 1619 . . . 4
23 cbvmpt2x.6 . . . . 5
2423nfeq2 2500 . . . 4
2522, 24nfan 1824 . . 3
26 eleq1 2413 . . . . . 6
2726adantr 451 . . . . 5
28 cbvmpt2x.7 . . . . . . 7
2928eleq2d 2420 . . . . . 6
30 eleq1 2413 . . . . . 6
3129, 30sylan9bb 680 . . . . 5
3227, 31anbi12d 691 . . . 4
33 cbvmpt2x.8 . . . . 5
3433eqeq2d 2364 . . . 4
3532, 34anbi12d 691 . . 3
367, 14, 21, 25, 35cbvoprab12 5569 . 2
37 df-mpt2 5654 . 2
38 df-mpt2 5654 . 2
3936, 37, 383eqtr4i 2383 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 176   wa 358   wceq 1642   wcel 1710  wnfc 2476  coprab 5527   cmpt2 5653 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-opab 4623  df-oprab 5528  df-mpt2 5654 This theorem is referenced by:  cbvmpt2  5679
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