Theorem necon4ai 2991

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 143	. 2 ⊢ (𝜑 → ¬ ¬ 𝜑)
2		necon4ai.1	. . 3 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝜑)
3	2	necon1bi 2988	. 2 ⊢ (¬ ¬ 𝜑 → 𝐴 = 𝐵)
4	1, 3	syl 18	1 ⊢ (𝜑 → 𝐴 = 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570   ≠ wne 2960

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

This proof depends on definitions:  df-bi 210  df-ne 2961

This theorem is used by:  necon4i  2995  dmsn0el  6214  funsneqopb  7153  sdom1  9213  cfeq0  10251