Theorem mteqand 3039

Description: A modus tollens deduction for inequality. (Contributed by Steven Nguyen, 1-Jun-2023.)

Hypotheses

Ref	Expression
mteqand.1	⊢ (𝜑 → 𝐶 ≠ 𝐷)
mteqand.2	⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐶 = 𝐷)

Assertion

Ref	Expression
mteqand	⊢ (𝜑 → 𝐴 ≠ 𝐵)

Proof of Theorem mteqand

Step	Hyp	Ref	Expression
1		mteqand.1	. . . 4 ⊢ (𝜑 → 𝐶 ≠ 𝐷)
2	1	neneqd 2951	. . 3 ⊢ (𝜑 → ¬ 𝐶 = 𝐷)
3		mteqand.2	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐶 = 𝐷)
4	2, 3	mtand 815	. 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐵)
5	4	neqned 2953	1 ⊢ (𝜑 → 𝐴 ≠ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ≠ wne 2946

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 207  df-an 396  df-ne 2947

This theorem is referenced by:  isdrngd  20787  imadrhmcl  20820  fracfld  33275  qsidomlem2  33446  rprmasso  33518  rtelextdg2lem  33717  2sqr3minply  33738  zarcmplem  33827  expeq1d  42311  remul01  42383  remulinvcom  42408  ricdrng1  42483  prjspersym  42562  prjspreln0  42564  prjspner1  42581  flt0  42592  fltne  42599