Theorem tfrlem3-2d 6583

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 5695	. . . . . 6 |- ( x = g -> ( F ` x ) = ( F ` g ) )
3	2	eleq1d 2307	. . . . 5 |- ( x = g -> ( ( F ` x ) e. _V <-> ( F ` g ) e. _V ) )
4	3	anbi2d 468	. . . 4 |- ( x = g -> ( ( Fun F /\ ( F ` x ) e. _V ) <-> ( Fun F /\ ( F ` g ) e. _V ) ) )
5	4	cbvalv 1973	. . 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 1611	1 |- ( ph -> ( Fun F /\ ( F ` g ) e. _V ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   A.wal 1400   e. wcel 2209   _Vcvv 2821   Fun wfun 5371   ` cfv 5377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-un 3224  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-iota 5337  df-fv 5385

This theorem is used by:  tfrlemisucfn  6595  tfrlemisucaccv  6596  tfrlemibxssdm  6598  tfrlemibfn  6599  tfrlemi14d  6604