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Theorem cbv1h 1675
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-May-2018.)
Hypotheses
Ref Expression
cbv1h.1  |-  ( ph  ->  ( ps  ->  A. y ps ) )
cbv1h.2  |-  ( ph  ->  ( ch  ->  A. x ch ) )
cbv1h.3  |-  ( ph  ->  ( x  =  y  ->  ( ps  ->  ch ) ) )
Assertion
Ref Expression
cbv1h  |-  ( A. x A. y ph  ->  ( A. x ps  ->  A. y ch ) )

Proof of Theorem cbv1h
StepHypRef Expression
1 nfa1 1475 . 2  |-  F/ x A. x A. y ph
2 nfa2 1512 . 2  |-  F/ y A. x A. y ph
3 sp 1442 . . . . 5  |-  ( A. y ph  ->  ph )
43sps 1471 . . . 4  |-  ( A. x A. y ph  ->  ph )
5 cbv1h.1 . . . 4  |-  ( ph  ->  ( ps  ->  A. y ps ) )
64, 5syl 14 . . 3  |-  ( A. x A. y ph  ->  ( ps  ->  A. y ps ) )
72, 6nfd 1457 . 2  |-  ( A. x A. y ph  ->  F/ y ps )
8 cbv1h.2 . . . 4  |-  ( ph  ->  ( ch  ->  A. x ch ) )
94, 8syl 14 . . 3  |-  ( A. x A. y ph  ->  ( ch  ->  A. x ch ) )
101, 9nfd 1457 . 2  |-  ( A. x A. y ph  ->  F/ x ch )
11 cbv1h.3 . . 3  |-  ( ph  ->  ( x  =  y  ->  ( ps  ->  ch ) ) )
124, 11syl 14 . 2  |-  ( A. x A. y ph  ->  ( x  =  y  -> 
( ps  ->  ch ) ) )
131, 2, 7, 10, 12cbv1 1674 1  |-  ( A. x A. y ph  ->  ( A. x ps  ->  A. y ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1283
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391
This theorem is referenced by:  cbv2h  1676
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