Theorem pm2.61da2ne 2595

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 695	. . 3 ⊢ (((φ ∧ A ≠ B) ∧ C = D) → ψ)
4		pm2.61da2ne.3	. . . 4 ⊢ ((φ ∧ (A ≠ B ∧ C ≠ D)) → ψ)
5	4	anassrs 629	. . 3 ⊢ (((φ ∧ A ≠ B) ∧ C ≠ D) → ψ)
6	3, 5	pm2.61dane 2594	. 2 ⊢ ((φ ∧ A ≠ B) → ψ)
7	1, 6	pm2.61dane 2594	1 ⊢ (φ → ψ)

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ≠ wne 2516

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 177  df-an 360  df-ne 2518

This theorem is referenced by:  pm2.61da3ne  2596