Theorem tfrlem3-2d 6316

Description: Lemma for transfinite recursion which changes a bound variable (Contributed by Jim Kingdon, 2-Jul-2019.)

Hypothesis

Ref	Expression
tfrlem3-2d.1	|- ( ph -> A. x ( Fun F /\ ( F ` x ) e. _V ) )

Assertion

Ref	Expression
tfrlem3-2d	|- ( ph -> ( Fun F /\ ( F ` g ) e. _V ) )

Distinct variable group:   x, g, F

Allowed substitution hints:   ph( x, g)

Proof of Theorem tfrlem3-2d

Step	Hyp	Ref	Expression
1		tfrlem3-2d.1	. . 3 |- ( ph -> A. x ( Fun F /\ ( F ` x ) e. _V ) )
2		fveq2 5517	. . . . . 6 |- ( x = g -> ( F ` x ) = ( F ` g ) )
3	2	eleq1d 2246	. . . . 5 |- ( x = g -> ( ( F ` x ) e. _V <-> ( F ` g ) e. _V ) )
4	3	anbi2d 464	. . . 4 |- ( x = g -> ( ( Fun F /\ ( F ` x ) e. _V ) <-> ( Fun F /\ ( F ` g ) e. _V ) ) )
5	4	cbvalv 1917	. . 3 |- ( A. x ( Fun F /\ ( F ` x ) e. _V ) <-> A. g ( Fun F /\ ( F ` g ) e. _V ) )
6	1, 5	sylib 122	. 2 |- ( ph -> A. g ( Fun F /\ ( F ` g ) e. _V ) )
7	6	19.21bi 1558	1 |- ( ph -> ( Fun F /\ ( F ` g ) e. _V ) )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1351   e. wcel 2148   _Vcvv 2739   Fun wfun 5212   ` cfv 5218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-iota 5180  df-fv 5226

This theorem is referenced by:  tfrlemisucfn  6328  tfrlemisucaccv  6329  tfrlemibxssdm  6331  tfrlemibfn  6332  tfrlemi14d  6337