Theorem pm2.61da2ne 3044

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 728	. . 3 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ 𝐶 = 𝐷) → 𝜓)
4		pm2.61da2ne.3	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷)) → 𝜓)
5	4	anassrs 473	. . 3 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ 𝐶 ≠ 𝐷) → 𝜓)
6	3, 5	pm2.61dane 3043	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝜓)
7	1, 6	pm2.61dane 3043	1 ⊢ (𝜑 → 𝜓)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ≠ wne 2956

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-ne 2957

This theorem is used by:  pm2.61da3ne  3045  isabvd  21069  xrsxmet  25129  chordthmlem3  27162  mumul  27508  lgsdirnn0  27671  lgsdinn0  27672  constrrtcc  34367  lfl1dim  40178  lfl1dim2N  40179  pmodlem2  40904  cdlemg29  41762  cdlemg39  41773  cdlemg44b  41789  dia2dimlem9  42129  dihprrn  42483  dvh3dim  42503  lcfl9a  42562  lclkrlem2l  42575  lcfrlem42  42641  mapdh6kN  42803  hdmap1l6k  42877