Theorem necon1bi 1652

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 1630	. . 3 |- (A =/= B <-> -. A = B)
2		necon1bi.1	. . 3 |- (A =/= B -> ph)
3	1, 2	sylbir 199	. 2 |- (-. A = B -> ph)
4	3	con1i 96	1 |- (-. ph -> A = B)

Syntax hints:  -. wn 2   -> wi 3   = wceq 992   =/= wne 1628

This theorem is referenced by:  peano5 3241  1st2val 4158  2nd2val 4159  eceqopreq 4454  mapprc 4467  pw2en 4587  setind 4794  isumnul 7407  hatomistici 10570

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 145  df-ne 1630

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