Theorem necon4ai 1624

Description: Contrapositive inference for inequality.

Hypothesis

Ref	Expression
necon4ai.1	|- (A =/= B -> -. ph)

Assertion

Ref	Expression
necon4ai	|- (ph -> A = B)

Proof of Theorem necon4ai

Step	Hyp	Ref	Expression
1		df-ne 1587	. . 3 |- (A =/= B <-> -. A = B)
2		necon4ai.1	. . 3 |- (A =/= B -> -. ph)
3	1, 2	sylbir 201	. 2 |- (-. A = B -> -. ph)
4	3	a3i 74	1 |- (ph -> A = B)

Syntax hints:  -. wn 2   -> wi 3   = wceq 956   =/= wne 1585

This theorem is referenced by:  necon4i 1625  cfeq0 4914

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 147  df-ne 1587

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