Theorem acongeq12d 43516

Description: Substitution deduction for alternating congruence. (Contributed by Stefan O'Rear, 3-Oct-2014.)

Hypotheses

Ref	Expression
acongeq12d.1	⊢ (𝜑 → 𝐵 = 𝐶)
acongeq12d.2	⊢ (𝜑 → 𝐷 = 𝐸)

Assertion

Ref	Expression
acongeq12d	⊢ (𝜑 → ((𝐴 ∥ (𝐵 − 𝐷) ∨ 𝐴 ∥ (𝐵 − -𝐷)) ↔ (𝐴 ∥ (𝐶 − 𝐸) ∨ 𝐴 ∥ (𝐶 − -𝐸))))

Proof of Theorem acongeq12d

Step	Hyp	Ref	Expression
1		acongeq12d.1	. . . 4 ⊢ (𝜑 → 𝐵 = 𝐶)
2		acongeq12d.2	. . . 4 ⊢ (𝜑 → 𝐷 = 𝐸)
3	1, 2	oveq12d 7408	. . 3 ⊢ (𝜑 → (𝐵 − 𝐷) = (𝐶 − 𝐸))
4	3	breq2d 5109	. 2 ⊢ (𝜑 → (𝐴 ∥ (𝐵 − 𝐷) ↔ 𝐴 ∥ (𝐶 − 𝐸)))
5	2	negeqd 11417	. . . 4 ⊢ (𝜑 → -𝐷 = -𝐸)
6	1, 5	oveq12d 7408	. . 3 ⊢ (𝜑 → (𝐵 − -𝐷) = (𝐶 − -𝐸))
7	6	breq2d 5109	. 2 ⊢ (𝜑 → (𝐴 ∥ (𝐵 − -𝐷) ↔ 𝐴 ∥ (𝐶 − -𝐸)))
8	4, 7	orbi12d 929	1 ⊢ (𝜑 → ((𝐴 ∥ (𝐵 − 𝐷) ∨ 𝐴 ∥ (𝐵 − -𝐷)) ↔ (𝐴 ∥ (𝐶 − 𝐸) ∨ 𝐴 ∥ (𝐶 − -𝐸))))

Syntax hints:   → wi 4   ↔ wb 208   ∨ wo 858   = wceq 1559   class class class wbr 5097  (class class class)co 7390   − cmin 11407  -cneg 11408   ∥ cdvds 16276

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-iota 6471  df-fv 6523  df-ov 7393  df-neg 11410

This theorem is referenced by:  acongrep  43517  jm2.26a  43537  jm2.26  43539