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Theorem pm2.61ne 2591
 Description: Deduction eliminating an inequality in an antecedent. (Contributed by NM, 24-May-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Hypotheses
Ref Expression
pm2.61ne.1 (A = B → (ψχ))
pm2.61ne.2 ((φ AB) → ψ)
pm2.61ne.3 (φχ)
Assertion
Ref Expression
pm2.61ne (φψ)

Proof of Theorem pm2.61ne
StepHypRef Expression
1 pm2.61ne.2 . . 3 ((φ AB) → ψ)
21expcom 424 . 2 (AB → (φψ))
3 nne 2520 . . 3 ABA = B)
4 pm2.61ne.3 . . . 4 (φχ)
5 pm2.61ne.1 . . . 4 (A = B → (ψχ))
64, 5syl5ibr 212 . . 3 (A = B → (φψ))
73, 6sylbi 187 . 2 AB → (φψ))
82, 7pm2.61i 156 1 (φψ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ 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: (None)
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