Theorem ifnefalse 3671

Description: When values are unequal, but an "if" condition checks if they are equal, then the "false" branch results. This is a simple utility to provide a slight shortening and simplification of proofs vs. applying iffalse 3670 directly in this case. It happens, e.g., in oevn0 in set.mm. (Contributed by David A. Wheeler, 15-May-2015.)

Assertion

Ref	Expression
ifnefalse	⊢ (A ≠ B → if(A = B, C, D) = D)

Proof of Theorem ifnefalse

Step	Hyp	Ref	Expression
1		df-ne 2519	. 2 ⊢ (A ≠ B ↔ ¬ A = B)
2		iffalse 3670	. 2 ⊢ (¬ A = B → if(A = B, C, D) = D)
3	1, 2	sylbi 187	1 ⊢ (A ≠ B → if(A = B, C, D) = D)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1642   ≠ wne 2517   ifcif 3663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-ne 2519  df-if 3664

This theorem is referenced by: (None)