Theorem necon2bi 2475

Description: Contrapositive inference for inequality. (Contributed by NM, 1-Apr-2007.)

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	1	neneqd 2441	. 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐵)
3	2	con2i 636	1 ⊢ (𝐴 = 𝐵 → ¬ 𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1402   ≠ wne 2420

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-in1 623  ax-in2 624

This proof depends on definitions:  df-bi 117  df-ne 2421

This theorem is used by:  minel  3586  rzal  3625  difsnb  3858  fin0  7189  0npi  7680  0nsr  8116  renfdisj  8385  nltpnft  10218  ngtmnft  10221  xrrebnd  10223  hashnncl  11236  rennim  11770  pceq0  13103