Theorem ifeq2da 15694

Description: Conditional equality.

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 2989	. . . 4 |- (ps -> if(ps, C, A) = C)
2		iftrue 2989	. . . 4 |- (ps -> if(ps, C, B) = C)
3	1, 2	eqtr4d 1928	. . 3 |- (ps -> if(ps, C, A) = if(ps, C, B))
4	3	adantl 424	. 2 |- ((ph /\ ps) -> if(ps, C, A) = if(ps, C, B))
5		ifeq2da.1	. . 3 |- ((ph /\ -. ps) -> A = B)
6	5	ifeq2d 2994	. 2 |- ((ph /\ -. ps) -> if(ps, C, A) = if(ps, C, B))
7	4, 6	pm2.61dan 535	1 |- (ph -> if(ps, C, A) = if(ps, C, B))

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

This theorem is referenced by:  pcoass 16085

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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