Theorem acongeq12d 43099

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 7372	. . 3 ⊢ (𝜑 → (𝐵 − 𝐷) = (𝐶 − 𝐸))
4	3	breq2d 5107	. 2 ⊢ (𝜑 → (𝐴 ∥ (𝐵 − 𝐷) ↔ 𝐴 ∥ (𝐶 − 𝐸)))
5	2	negeqd 11363	. . . 4 ⊢ (𝜑 → -𝐷 = -𝐸)
6	1, 5	oveq12d 7372	. . 3 ⊢ (𝜑 → (𝐵 − -𝐷) = (𝐶 − -𝐸))
7	6	breq2d 5107	. 2 ⊢ (𝜑 → (𝐴 ∥ (𝐵 − -𝐷) ↔ 𝐴 ∥ (𝐶 − -𝐸)))
8	4, 7	orbi12d 918	1 ⊢ (𝜑 → ((𝐴 ∥ (𝐵 − 𝐷) ∨ 𝐴 ∥ (𝐵 − -𝐷)) ↔ (𝐴 ∥ (𝐶 − 𝐸) ∨ 𝐴 ∥ (𝐶 − -𝐸))))

Syntax hints:   → wi 4   ↔ wb 206   ∨ wo 847   = wceq 1541   class class class wbr 5095  (class class class)co 7354   − cmin 11353  -cneg 11354   ∥ cdvds 16167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-iota 6444  df-fv 6496  df-ov 7357  df-neg 11356

This theorem is referenced by:  acongrep  43100  jm2.26a  43120  jm2.26  43122