Theorem mteqand 3047

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 2961	. . 3 ⊢ (𝜑 → ¬ 𝐶 = 𝐷)
3		mteqand.2	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐶 = 𝐷)
4	2, 3	mtand 825	. 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐵)
5	4	neqned 2963	1 ⊢ (𝜑 → 𝐴 ≠ 𝐵)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ≠ wne 2956

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 209  df-an 400  df-ne 2957

This theorem is referenced by:  isdrngd  20794  imadrhmcl  20826  fracfld  33456  qsidomlem2  33601  rprmasso  33682  vr1nz  33750  rtelextdg2lem  33984  2sqr3minply  34038  cos9thpiminplylem2  34041  zarcmplem  34139  expeq1d  42897  remul01  42980  remulinvcom  43006  mulgt0b2d  43064  sn-inelr  43073  ricdrng1  43110  prjspersym  43153  prjspreln0  43155  prjspner1  43172  flt0  43183  fltne  43190  eufunc  50107