Theorem pm2.61dane 2594

Description: Deduction eliminating an inequality in an antecedent. (Contributed by NM, 30-Nov-2011.)

Hypotheses

Ref	Expression
pm2.61dane.1	⊢ ((φ ∧ A = B) → ψ)
pm2.61dane.2	⊢ ((φ ∧ A ≠ B) → ψ)

Assertion

Ref	Expression
pm2.61dane	⊢ (φ → ψ)

Proof of Theorem pm2.61dane

Step	Hyp	Ref	Expression
1		pm2.61dane.1	. . 3 ⊢ ((φ ∧ A = B) → ψ)
2	1	ex 423	. 2 ⊢ (φ → (A = B → ψ))
3		pm2.61dane.2	. . 3 ⊢ ((φ ∧ A ≠ B) → ψ)
4	3	ex 423	. 2 ⊢ (φ → (A ≠ B → ψ))
5	2, 4	pm2.61dne 2593	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:  pm2.61da2ne  2595  pm2.61da3ne  2596