Theorem pm2.61da3ne 3034

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	⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝜓)
pm2.61da3ne.2	⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝜓)
pm2.61da3ne.3	⊢ ((𝜑 ∧ 𝐸 = 𝐹) → 𝜓)
pm2.61da3ne.4	⊢ ((𝜑 ∧ (𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷 ∧ 𝐸 ≠ 𝐹)) → 𝜓)

Assertion

Ref	Expression
pm2.61da3ne	⊢ (𝜑 → 𝜓)

Proof of Theorem pm2.61da3ne

Step	Hyp	Ref	Expression
1		pm2.61da3ne.2	. 2 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝜓)
2		pm2.61da3ne.3	. 2 ⊢ ((𝜑 ∧ 𝐸 = 𝐹) → 𝜓)
3		pm2.61da3ne.1	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝜓)
4	3	a1d 25	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → ((𝐶 ≠ 𝐷 ∧ 𝐸 ≠ 𝐹) → 𝜓))
5		pm2.61da3ne.4	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷 ∧ 𝐸 ≠ 𝐹)) → 𝜓)
6	5	3exp2 1353	. . . . 5 ⊢ (𝜑 → (𝐴 ≠ 𝐵 → (𝐶 ≠ 𝐷 → (𝐸 ≠ 𝐹 → 𝜓))))
7	6	imp4b 422	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → ((𝐶 ≠ 𝐷 ∧ 𝐸 ≠ 𝐹) → 𝜓))
8	4, 7	pm2.61dane 3032	. . 3 ⊢ (𝜑 → ((𝐶 ≠ 𝐷 ∧ 𝐸 ≠ 𝐹) → 𝜓))
9	8	imp 407	. 2 ⊢ ((𝜑 ∧ (𝐶 ≠ 𝐷 ∧ 𝐸 ≠ 𝐹)) → 𝜓)
10	1, 2, 9	pm2.61da2ne 3033	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ≠ wne 2943

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 206  df-an 397  df-3an 1088  df-ne 2944

This theorem is referenced by:  trljco  38754  dvh4dimN  39461