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Theorem cbvrab 2857
 Description: Rule to change the bound variable in a restricted class abstraction, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 9-Oct-2016.)
Hypotheses
Ref Expression
cbvrab.1
cbvrab.2
cbvrab.3
cbvrab.4
cbvrab.5
Assertion
Ref Expression
cbvrab

Proof of Theorem cbvrab
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfv 1619 . . . 4
2 cbvrab.1 . . . . . 6
32nfcri 2483 . . . . 5
4 nfs1v 2106 . . . . 5
53, 4nfan 1824 . . . 4
6 eleq1 2413 . . . . 5
7 sbequ12 1919 . . . . 5
86, 7anbi12d 691 . . . 4
91, 5, 8cbvab 2471 . . 3
10 cbvrab.2 . . . . . 6
1110nfcri 2483 . . . . 5
12 cbvrab.3 . . . . . 6
1312nfsb 2109 . . . . 5
1411, 13nfan 1824 . . . 4
15 nfv 1619 . . . 4
16 eleq1 2413 . . . . 5
17 sbequ 2060 . . . . . 6
18 cbvrab.4 . . . . . . 7
19 cbvrab.5 . . . . . . 7
2018, 19sbie 2038 . . . . . 6
2117, 20syl6bb 252 . . . . 5
2216, 21anbi12d 691 . . . 4
2314, 15, 22cbvab 2471 . . 3
249, 23eqtri 2373 . 2
25 df-rab 2623 . 2
26 df-rab 2623 . 2
2724, 25, 263eqtr4i 2383 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 176   wa 358  wnf 1544   wceq 1642  wsb 1648   wcel 1710  cab 2339  wnfc 2476  crab 2618 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-rab 2623 This theorem is referenced by:  cbvrabv  2858  elrabsf  3084
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