Theorem necon4ai 3050

Description: Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 22-Nov-2019.)

Hypothesis

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

Assertion

Ref	Expression
necon4ai	⊢ (𝜑 → 𝐴 = 𝐵)

Proof of Theorem necon4ai

Step	Hyp	Ref	Expression
1		notnot 144	. 2 ⊢ (𝜑 → ¬ ¬ 𝜑)
2		necon4ai.1	. . 3 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝜑)
3	2	necon1bi 3047	. 2 ⊢ (¬ ¬ 𝜑 → 𝐴 = 𝐵)
4	1, 3	syl 17	1 ⊢ (𝜑 → 𝐴 = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1536   ≠ wne 3019

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 209  df-ne 3020

This theorem is referenced by:  necon4i  3054  dmsn0el  6071  funsneqopb  6917  cfeq0  9681