Theorem pm2.61da2ne 3107

Description: Deduction eliminating two inequalities in an antecedent. (Contributed by NM, 29-May-2013.)

Hypotheses

Ref	Expression
pm2.61da2ne.1	⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝜓)
pm2.61da2ne.2	⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝜓)
pm2.61da2ne.3	⊢ ((𝜑 ∧ (𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷)) → 𝜓)

Assertion

Ref	Expression
pm2.61da2ne	⊢ (𝜑 → 𝜓)

Proof of Theorem pm2.61da2ne

Step	Hyp	Ref	Expression
1		pm2.61da2ne.1	. 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝜓)
2		pm2.61da2ne.2	. . . 4 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝜓)
3	2	adantlr 713	. . 3 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ 𝐶 = 𝐷) → 𝜓)
4		pm2.61da2ne.3	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷)) → 𝜓)
5	4	anassrs 470	. . 3 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ 𝐶 ≠ 𝐷) → 𝜓)
6	3, 5	pm2.61dane 3106	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝜓)
7	1, 6	pm2.61dane 3106	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ≠ wne 3018

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 399  df-ne 3019

This theorem is referenced by:  pm2.61da3ne  3108  isabvd  19593  xrsxmet  23419  chordthmlem3  25414  mumul  25760  lgsdirnn0  25922  lgsdinn0  25923  lfl1dim  36259  lfl1dim2N  36260  pmodlem2  36985  cdlemg29  37843  cdlemg39  37854  cdlemg44b  37870  dia2dimlem9  38210  dihprrn  38564  dvh3dim  38584  lcfl9a  38643  lclkrlem2l  38656  lcfrlem42  38722  mapdh6kN  38884  hdmap1l6k  38958