Theorem necon2i 2993

Description: Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.)

Hypothesis

Ref	Expression
necon2i.1	⊢ (𝐴 = 𝐵 → 𝐶 ≠ 𝐷)

Assertion

Ref	Expression
necon2i	⊢ (𝐶 = 𝐷 → 𝐴 ≠ 𝐵)

Proof of Theorem necon2i

Step	Hyp	Ref	Expression
1		necon2i.1	. . 3 ⊢ (𝐴 = 𝐵 → 𝐶 ≠ 𝐷)
2	1	neneqd 2964	. 2 ⊢ (𝐴 = 𝐵 → ¬ 𝐶 = 𝐷)
3	2	necon2ai 2988	1 ⊢ (𝐶 = 𝐷 → 𝐴 ≠ 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ≠ wne 2959

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 2960

This theorem is used by:  cmpfi  23759  mcubic  27208  cubic2  27209  2sqlem11  27789  zarcmplem  34596  ovoliunnfl  38785  voliunnfl  38787  volsupnfl  38788  mncn0  44344  aaitgo  44367  usgrexmpl2trifr  49350