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Theorem pm2.61da3ne 3079
Description: Deduction eliminating three inequalities in an antecedent. (Contributed by NM, 15-Jun-2013.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
Hypotheses
Ref Expression
pm2.61da3ne.1 ((𝜑𝐴 = 𝐵) → 𝜓)
pm2.61da3ne.2 ((𝜑𝐶 = 𝐷) → 𝜓)
pm2.61da3ne.3 ((𝜑𝐸 = 𝐹) → 𝜓)
pm2.61da3ne.4 ((𝜑 ∧ (𝐴𝐵𝐶𝐷𝐸𝐹)) → 𝜓)
Assertion
Ref Expression
pm2.61da3ne (𝜑𝜓)

Proof of Theorem pm2.61da3ne
StepHypRef Expression
1 pm2.61da3ne.2 . 2 ((𝜑𝐶 = 𝐷) → 𝜓)
2 pm2.61da3ne.3 . 2 ((𝜑𝐸 = 𝐹) → 𝜓)
3 pm2.61da3ne.1 . . . . 5 ((𝜑𝐴 = 𝐵) → 𝜓)
43a1d 25 . . . 4 ((𝜑𝐴 = 𝐵) → ((𝐶𝐷𝐸𝐹) → 𝜓))
5 pm2.61da3ne.4 . . . . . 6 ((𝜑 ∧ (𝐴𝐵𝐶𝐷𝐸𝐹)) → 𝜓)
653exp2 1351 . . . . 5 (𝜑 → (𝐴𝐵 → (𝐶𝐷 → (𝐸𝐹𝜓))))
76imp4b 425 . . . 4 ((𝜑𝐴𝐵) → ((𝐶𝐷𝐸𝐹) → 𝜓))
84, 7pm2.61dane 3077 . . 3 (𝜑 → ((𝐶𝐷𝐸𝐹) → 𝜓))
98imp 410 . 2 ((𝜑 ∧ (𝐶𝐷𝐸𝐹)) → 𝜓)
101, 2, 9pm2.61da2ne 3078 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wne 2990
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 210  df-an 400  df-3an 1086  df-ne 2991
This theorem is referenced by:  trljco  38035  dvh4dimN  38742
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