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Theorem cbvralf 2829
 Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 7-Mar-2004.) (Revised by Mario Carneiro, 9-Oct-2016.)
Hypotheses
Ref Expression
cbvralf.1
cbvralf.2
cbvralf.3
cbvralf.4
cbvralf.5
Assertion
Ref Expression
cbvralf

Proof of Theorem cbvralf
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfv 1619 . . . 4
2 cbvralf.1 . . . . . 6
32nfcri 2483 . . . . 5
4 nfs1v 2106 . . . . 5
53, 4nfim 1813 . . . 4
6 eleq1 2413 . . . . 5
7 sbequ12 1919 . . . . 5
86, 7imbi12d 311 . . . 4
91, 5, 8cbval 1984 . . 3
10 cbvralf.2 . . . . . 6
1110nfcri 2483 . . . . 5
12 cbvralf.3 . . . . . 6
1312nfsb 2109 . . . . 5
1411, 13nfim 1813 . . . 4
15 nfv 1619 . . . 4
16 eleq1 2413 . . . . 5
17 sbequ 2060 . . . . . 6
18 cbvralf.4 . . . . . . 7
19 cbvralf.5 . . . . . . 7
2018, 19sbie 2038 . . . . . 6
2117, 20syl6bb 252 . . . . 5
2216, 21imbi12d 311 . . . 4
2314, 15, 22cbval 1984 . . 3
249, 23bitri 240 . 2
25 df-ral 2619 . 2
26 df-ral 2619 . 2
2724, 25, 263bitr4i 268 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 176  wal 1540  wnf 1544   wceq 1642  wsb 1648   wcel 1710  wnfc 2476  wral 2614 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-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619 This theorem is referenced by:  cbvrexf  2830  cbvral  2831  ffnfvf  5428
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