Theorem necon1bbid 3007

Description: Contrapositive inference for inequality. (Contributed by NM, 31-Jan-2008.)

Hypothesis

Ref	Expression
necon1bbid.1	⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ 𝜓))

Assertion

Ref	Expression
necon1bbid	⊢ (𝜑 → (¬ 𝜓 ↔ 𝐴 = 𝐵))

Proof of Theorem necon1bbid

Step	Hyp	Ref	Expression
1		df-ne 2969	. . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
2		necon1bbid.1	. . 3 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ 𝜓))
3	1, 2	syl5bbr 277	. 2 ⊢ (𝜑 → (¬ 𝐴 = 𝐵 ↔ 𝜓))
4	3	con1bid 347	1 ⊢ (𝜑 → (¬ 𝜓 ↔ 𝐴 = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   = wceq 1601   ≠ wne 2968

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 199  df-ne 2969

This theorem is referenced by:  necon4abid  3008  blssioo  23006  metdstri  23062  rrxmvallem  23610  dchrpt  25444  lgsquad3  25564  eupth2lem2  27637  lkrpssN  35311  dochshpsat  37602