Theorem necon4ai 2576

Description: Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypothesis

Ref	Expression
necon4ai.1	⊢ (A ≠ B → ¬ φ)

Assertion

Ref	Expression
necon4ai	⊢ (φ → A = B)

Proof of Theorem necon4ai

Step	Hyp	Ref	Expression
1		necon4ai.1	. . 3 ⊢ (A ≠ B → ¬ φ)
2	1	con2i 112	. 2 ⊢ (φ → ¬ A ≠ B)
3		nne 2521	. 2 ⊢ (¬ A ≠ B ↔ A = B)
4	2, 3	sylib 188	1 ⊢ (φ → A = B)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1642   ≠ wne 2517

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-ne 2519

This theorem is used by: (None)