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  20745  xrsxmet  24754  chordthmlem3  26800  mumul  27147  lgsdirnn0  27311  lgsdinn0  27312  constrrtcc  33892  lfl1dim  39377  lfl1dim2N  39378  pmodlem2  40103  cdlemg29  40961  cdlemg39  40972  cdlemg44b  40988  dia2dimlem9  41328  dihprrn  41682  dvh3dim  41702  lcfl9a  41761  lclkrlem2l  41774  lcfrlem42  41840  mapdh6kN  42002  hdmap1l6k  42076