Theorem pm2.61iine 3023

Description: Equality version of pm2.61ii 183. (Contributed by Scott Fenton, 13-Jun-2013.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)

Hypotheses

Ref	Expression
pm2.61iine.1	⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) → 𝜑)
pm2.61iine.2	⊢ (𝐴 = 𝐶 → 𝜑)
pm2.61iine.3	⊢ (𝐵 = 𝐷 → 𝜑)

Assertion

Ref	Expression
pm2.61iine	⊢ 𝜑

Proof of Theorem pm2.61iine

Step	Hyp	Ref	Expression
1		pm2.61iine.2	. 2 ⊢ (𝐴 = 𝐶 → 𝜑)
2		pm2.61iine.3	. . . 4 ⊢ (𝐵 = 𝐷 → 𝜑)
3	2	adantl 481	. . 3 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 = 𝐷) → 𝜑)
4		pm2.61iine.1	. . 3 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) → 𝜑)
5	3, 4	pm2.61dane 3020	. 2 ⊢ (𝐴 ≠ 𝐶 → 𝜑)
6	1, 5	pm2.61ine 3016	1 ⊢ 𝜑

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ≠ wne 2933

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 207  df-an 396  df-ne 2934

This theorem is referenced by:  fntpb  7206  elfiun  9447  dedekind  11403  mdsymi  32397  isubgr3stgrlem6  47950