Theorem pm2.61da2ne 3016

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 715	. . 3 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ 𝐶 = 𝐷) → 𝜓)
4		pm2.61da2ne.3	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷)) → 𝜓)
5	4	anassrs 467	. . 3 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ 𝐶 ≠ 𝐷) → 𝜓)
6	3, 5	pm2.61dane 3015	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝜓)
7	1, 6	pm2.61dane 3015	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ≠ wne 2928

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 2929

This theorem is referenced by:  pm2.61da3ne  3017  isabvd  20727  xrsxmet  24725  chordthmlem3  26771  mumul  27118  lgsdirnn0  27282  lgsdinn0  27283  constrrtcc  33748  lfl1dim  39230  lfl1dim2N  39231  pmodlem2  39956  cdlemg29  40814  cdlemg39  40825  cdlemg44b  40841  dia2dimlem9  41181  dihprrn  41535  dvh3dim  41555  lcfl9a  41614  lclkrlem2l  41627  lcfrlem42  41693  mapdh6kN  41855  hdmap1l6k  41929