Theorem pm2.61da2ne 3020

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 3019	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝜓)
7	1, 6	pm2.61dane 3019	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ≠ wne 2932

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 2933

This theorem is referenced by:  pm2.61da3ne  3021  isabvd  20747  xrsxmet  24756  chordthmlem3  26802  mumul  27149  lgsdirnn0  27313  lgsdinn0  27314  constrrtcc  33894  lfl1dim  39403  lfl1dim2N  39404  pmodlem2  40129  cdlemg29  40987  cdlemg39  40998  cdlemg44b  41014  dia2dimlem9  41354  dihprrn  41708  dvh3dim  41728  lcfl9a  41787  lclkrlem2l  41800  lcfrlem42  41866  mapdh6kN  42028  hdmap1l6k  42102