Theorem necon1bi 1601

Description: Contrapositive inference for inequality.

Hypothesis

Ref	Expression
necon1bi.1	|- (A =/= B -> ph)

Assertion

Ref	Expression
necon1bi	|- (-. ph -> A = B)

Proof of Theorem necon1bi

Step	Hyp	Ref	Expression
1		df-ne 1579	. . 3 |- (A =/= B <-> -. A = B)
2		necon1bi.1	. . 3 |- (A =/= B -> ph)
3	1, 2	sylbir 201	. 2 |- (-. A = B -> ph)
4	3	con1i 96	1 |- (-. ph -> A = B)

Syntax hints:  -. wn 2   -> wi 3   = wceq 953   =/= wne 1577

This theorem is referenced by:  peano5 3143  mapprc 4310  pw2en 4426  setind 4620  isumnul 7138  hatomistic 10197

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 1579

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