Theorem ifeq2da 4117

Description: Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)

Hypothesis

Ref	Expression
ifeq2da.1	|- ( ( ph /\ -. ps ) -> A = B )

Assertion

Ref	Expression
ifeq2da	|- ( ph -> if ( ps , C , A ) = if ( ps , C , B ) )

Proof of Theorem ifeq2da

Step	Hyp	Ref	Expression
1		iftrue 4092	. . . 4 |- ( ps -> if ( ps , C , A ) = C )
2		iftrue 4092	. . . 4 |- ( ps -> if ( ps , C , B ) = C )
3	1, 2	eqtr4d 2659	. . 3 |- ( ps -> if ( ps , C , A ) = if ( ps , C , B ) )
4	3	adantl 482	. 2 |- ( ( ph /\ ps ) -> if ( ps , C , A ) = if ( ps , C , B ) )
5		ifeq2da.1	. . 3 |- ( ( ph /\ -. ps ) -> A = B )
6	5	ifeq2d 4105	. 2 |- ( ( ph /\ -. ps ) -> if ( ps , C , A ) = if ( ps , C , B ) )
7	4, 6	pm2.61dan 832	1 |- ( ph -> if ( ps , C , A ) = if ( ps , C , B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   ifcif 4086

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-nfc 2753  df-rab 2921  df-v 3202  df-un 3579  df-if 4087

This theorem is referenced by:  ifeq12da  4118  dfac12lem1  8965  ttukeylem3  9333  xmulcom  12096  xmulneg1  12099  subgmulg  17608  1marepvmarrepid  20381  copco  22818  pcopt2  22823  limcdif  23640  limcmpt  23647  limcres  23650  limccnp  23655  radcnv0  24170  leibpi  24669  efrlim  24696  dchrvmasumiflem2  25191  rpvmasum2  25201  padicabvf  25320  padicabvcxp  25321  itg2addnclem  33461  fourierdlem73  40396  fourierdlem76  40399  fourierdlem89  40412  fourierdlem91  40414