Theorem ifeq1da 15693

Description: Conditional equality.

Hypothesis

Ref	Expression
ifeq1da.1	|- ((ph /\ ps) -> A = B)

Assertion

Ref	Expression
ifeq1da	|- (ph -> if(ps, A, C) = if(ps, B, C))

Proof of Theorem ifeq1da

Step	Hyp	Ref	Expression
1		ifeq1da.1	. . 3 |- ((ph /\ ps) -> A = B)
2	1	ifeq1d 2993	. 2 |- ((ph /\ ps) -> if(ps, A, C) = if(ps, B, C))
3		iffalse 2991	. . . 4 |- (-. ps -> if(ps, A, C) = C)
4		iffalse 2991	. . . 4 |- (-. ps -> if(ps, B, C) = C)
5	3, 4	eqtr4d 1928	. . 3 |- (-. ps -> if(ps, A, C) = if(ps, B, C))
6	5	adantl 424	. 2 |- ((ph /\ -. ps) -> if(ps, A, C) = if(ps, B, C))
7	2, 6	pm2.61dan 535	1 |- (ph -> if(ps, A, C) = if(ps, B, C))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240   = wceq 1298  ifcif 2982

This theorem is referenced by:  pcopt 16084  pcoass 16085  pcorev 16087

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-10 1308  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-if 2983

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