Theorem ibllem 23531

Description: Conditioned equality theorem for the if statement. (Contributed by Mario Carneiro, 31-Jul-2014.)

Hypothesis

Ref	Expression
ibllem.1	|- ( ( ph /\ x e. A ) -> B = C )

Assertion

Ref	Expression
ibllem	|- ( ph -> if ( ( x e. A /\ 0 <_ B ) , B , 0 ) = if ( ( x e. A /\ 0 <_ C ) , C , 0 ) )

Proof of Theorem ibllem

Step	Hyp	Ref	Expression
1		ibllem.1	. . . . 5 |- ( ( ph /\ x e. A ) -> B = C )
2	1	breq2d 4665	. . . 4 |- ( ( ph /\ x e. A ) -> ( 0 <_ B <-> 0 <_ C ) )
3	2	pm5.32da 673	. . 3 |- ( ph -> ( ( x e. A /\ 0 <_ B ) <-> ( x e. A /\ 0 <_ C ) ) )
4	3	ifbid 4108	. 2 |- ( ph -> if ( ( x e. A /\ 0 <_ B ) , B , 0 ) = if ( ( x e. A /\ 0 <_ C ) , B , 0 ) )
5	1	adantrr 753	. . 3 |- ( ( ph /\ ( x e. A /\ 0 <_ C ) ) -> B = C )
6	5	ifeq1da 4116	. 2 |- ( ph -> if ( ( x e. A /\ 0 <_ C ) , B , 0 ) = if ( ( x e. A /\ 0 <_ C ) , C , 0 ) )
7	4, 6	eqtrd 2656	1 |- ( ph -> if ( ( x e. A /\ 0 <_ B ) , B , 0 ) = if ( ( x e. A /\ 0 <_ C ) , C , 0 ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   ifcif 4086   class class class wbr 4653   0cc0 9936   <_ cle 10075

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-3an 1039  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-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654

This theorem is referenced by:  isibl  23532  isibl2  23533  iblitg  23535  iblcnlem1  23554  iblcnlem  23555  itgcnlem  23556  iblrelem  23557  itgrevallem1  23561  itgeqa  23580