Theorem ifeq1 2364

Description: Equality theorem for conditional operator.

Assertion

Ref	Expression
ifeq1	|- (A = B -> if(ph, A, C) = if(ph, B, C))

Proof of Theorem ifeq1

Step	Hyp	Ref	Expression
1		eleq2 1535	. . . . 5 |- (A = B -> (x e. A <-> x e. B))
2	1	anbi1d 617	. . . 4 |- (A = B -> ((x e. A /\ ph) <-> (x e. B /\ ph)))
3	2	orbi1d 615	. . 3 |- (A = B -> (((x e. A /\ ph) \/ (x e. C /\ -. ph)) <-> ((x e. B /\ ph) \/ (x e. C /\ -. ph))))
4	3	abbidv 1577	. 2 |- (A = B -> {x | ((x e. A /\ ph) \/ (x e. C /\ -. ph))} = {x | ((x e. B /\ ph) \/ (x e. C /\ -. ph))})
5		df-if 2362	. 2 |- if(ph, A, C) = {x | ((x e. A /\ ph) \/ (x e. C /\ -. ph))}
6		df-if 2362	. 2 |- if(ph, B, C) = {x | ((x e. B /\ ph) \/ (x e. C /\ -. ph))}
7	4, 5, 6	3eqtr4g 1531	1 |- (A = B -> if(ph, A, C) = if(ph, B, C))

Syntax hints:  -. wn 2   -> wi 3   \/ wo 222   /\ wa 223   = wceq 956   e. wcel 958  {cab 1463  ifcif 2361

This theorem is referenced by:  ifeq12 2368  ifeq1d 2369  rdgeq2 3935

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-if 2362

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