Theorem ifid 2373

Description: Identical true and false arguments in the conditional operator.

Assertion

Ref	Expression
ifid	⊢ if(φ, A, A) = A

Proof of Theorem ifid

Step	Hyp	Ref	Expression
1		iftrue 2363	. 2 ⊢ (φ → if(φ, A, A) = A)
2		iffalse 2364	. 2 ⊢ (¬ φ → if(φ, A, A) = A)
3	1, 2	pm2.61i 126	1 ⊢ if(φ, A, A) = A

Syntax hints:   = wceq 955   ifcif 2358

This theorem is referenced by:  supsn 4574  metxptval 7792  metxp 7796

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-if 2359

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