Theorem necon2bi 1615

Description: Contrapositive inference for inequality.

Hypothesis

Ref	Expression
necon2bi.1	⊢ (φ → A ≠ B)

Assertion

Ref	Expression
necon2bi	⊢ (A = B → ¬ φ)

Proof of Theorem necon2bi

Step	Hyp	Ref	Expression
1		necon2bi.1	. . 3 ⊢ (φ → A ≠ B)
2		df-ne 1590	. . 3 ⊢ (A ≠ B ↔ ¬ A = B)
3	1, 2	sylib 198	. 2 ⊢ (φ → ¬ A = B)
4	3	con2i 97	1 ⊢ (A = B → ¬ φ)

Syntax hints:  ¬ wn 2   → wi 3   = wceq 958   ≠ wne 1588

This theorem is referenced by:  minel 2328  dtrucor2 2780  nlim0 3033  kmlem6 4780  0npi 5022  0npr 5108

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 1590

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