Theorem ifeq2 2361

Description: Equality theorem for conditional operator.

Assertion

Ref	Expression
ifeq2	|- (A = B -> if(ph, C, A) = if(ph, C, B))

Proof of Theorem ifeq2

Step	Hyp	Ref	Expression
1		eleq2 1532	. . . . 5 |- (A = B -> (x e. A <-> x e. B))
2	1	anbi1d 616	. . . 4 |- (A = B -> ((x e. A /\ -. ph) <-> (x e. B /\ -. ph)))
3	2	orbi2d 613	. . 3 |- (A = B -> (((x e. C /\ ph) \/ (x e. A /\ -. ph)) <-> ((x e. C /\ ph) \/ (x e. B /\ -. ph))))
4	3	abbidv 1574	. 2 |- (A = B -> {x | ((x e. C /\ ph) \/ (x e. A /\ -. ph))} = {x | ((x e. C /\ ph) \/ (x e. B /\ -. ph))})
5		df-if 2358	. 2 |- if(ph, C, A) = {x | ((x e. C /\ ph) \/ (x e. A /\ -. ph))}
6		df-if 2358	. 2 |- if(ph, C, B) = {x | ((x e. C /\ ph) \/ (x e. B /\ -. ph))}
7	4, 5, 6	3eqtr4g 1528	1 |- (A = B -> if(ph, C, A) = if(ph, C, B))

Syntax hints:  -. wn 2   -> wi 3   \/ wo 222   /\ wa 223   = wceq 954   e. wcel 956  {cab 1461  ifcif 2357

This theorem is referenced by:  ifeq12 2364  ifeq2d 2366  unxpdomlem 4823  metxptval 7782  spwval2 8595

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-if 2358

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