Theorem stoweidlem4 40221

Description: Lemma for stoweid 40280: a class variable replaces a setvar variable, for constant functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypothesis

Ref	Expression
stoweidlem4.1	|- ( ( ph /\ x e. RR ) -> ( t e. T |-> x ) e. A )

Assertion

Ref	Expression
stoweidlem4	|- ( ( ph /\ B e. RR ) -> ( t e. T |-> B ) e. A )

Distinct variable groups:   x, t, B   x, A   x, T   ph, x

Allowed substitution hints:   ph( t)   A( t)   T( t)

Proof of Theorem stoweidlem4

Step	Hyp	Ref	Expression
1		eleq1 2689	. . . . 5 |- ( x = B -> ( x e. RR <-> B e. RR ) )
2	1	anbi2d 740	. . . 4 |- ( x = B -> ( ( ph /\ x e. RR ) <-> ( ph /\ B e. RR ) ) )
3		simpl 473	. . . . . 6 |- ( ( x = B /\ t e. T ) -> x = B )
4	3	mpteq2dva 4744	. . . . 5 |- ( x = B -> ( t e. T |-> x ) = ( t e. T |-> B ) )
5	4	eleq1d 2686	. . . 4 |- ( x = B -> ( ( t e. T |-> x ) e. A <-> ( t e. T |-> B ) e. A ) )
6	2, 5	imbi12d 334	. . 3 |- ( x = B -> ( ( ( ph /\ x e. RR ) -> ( t e. T |-> x ) e. A ) <-> ( ( ph /\ B e. RR ) -> ( t e. T |-> B ) e. A ) ) )
7		stoweidlem4.1	. . 3 |- ( ( ph /\ x e. RR ) -> ( t e. T |-> x ) e. A )
8	6, 7	vtoclg 3266	. 2 |- ( B e. RR -> ( ( ph /\ B e. RR ) -> ( t e. T |-> B ) e. A ) )
9	8	anabsi7 860	1 |- ( ( ph /\ B e. RR ) -> ( t e. T |-> B ) e. A )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   |-> cmpt 4729   RRcr 9935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-ral 2917  df-v 3202  df-opab 4713  df-mpt 4730

This theorem is referenced by:  stoweidlem18  40235  stoweidlem19  40236  stoweidlem22  40239  stoweidlem32  40249  stoweidlem36  40253  stoweidlem40  40257  stoweidlem41  40258  stoweidlem55  40272