Theorem tfrlem3-2d 6217

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 5429	. . . . . 6 |- ( x = g -> ( F ` x ) = ( F ` g ) )
3	2	eleq1d 2209	. . . . 5 |- ( x = g -> ( ( F ` x ) e. _V <-> ( F ` g ) e. _V ) )
4	3	anbi2d 460	. . . 4 |- ( x = g -> ( ( Fun F /\ ( F ` x ) e. _V ) <-> ( Fun F /\ ( F ` g ) e. _V ) ) )
5	4	cbvalv 1890	. . 3 |- ( A. x ( Fun F /\ ( F ` x ) e. _V ) <-> A. g ( Fun F /\ ( F ` g ) e. _V ) )
6	1, 5	sylib 121	. 2 |- ( ph -> A. g ( Fun F /\ ( F ` g ) e. _V ) )
7	6	19.21bi 1538	1 |- ( ph -> ( Fun F /\ ( F ` g ) e. _V ) )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1330   e. wcel 1481   _Vcvv 2689   Fun wfun 5125   ` cfv 5131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-iota 5096  df-fv 5139

This theorem is referenced by:  tfrlemisucfn  6229  tfrlemisucaccv  6230  tfrlemibxssdm  6232  tfrlemibfn  6233  tfrlemi14d  6238