Theorem ifeqor 4132

Description: The possible values of a conditional operator. (Contributed by NM, 17-Jun-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
ifeqor	|- ( if ( ph , A , B ) = A \/ if ( ph , A , B ) = B )

Proof of Theorem ifeqor

Step	Hyp	Ref	Expression
1		iftrue 4092	. . . 4 |- ( ph -> if ( ph , A , B ) = A )
2	1	con3i 150	. . 3 |- ( -. if ( ph , A , B ) = A -> -. ph )
3	2	iffalsed 4097	. 2 |- ( -. if ( ph , A , B ) = A -> if ( ph , A , B ) = B )
4	3	orri 391	1 |- ( if ( ph , A , B ) = A \/ if ( ph , A , B ) = B )

Syntax hints:   -. wn 3   \/ wo 383   = wceq 1483   ifcif 4086

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-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-if 4087

This theorem is referenced by:  ifpr  4233  rabrsn  4259  prmolefac  15750  muval2  24860  finxpreclem2  33227  relexpxpmin  38009