Theorem acongeq12d 43333

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 7386	. . 3 ⊢ (𝜑 → (𝐵 − 𝐷) = (𝐶 − 𝐸))
4	3	breq2d 5112	. 2 ⊢ (𝜑 → (𝐴 ∥ (𝐵 − 𝐷) ↔ 𝐴 ∥ (𝐶 − 𝐸)))
5	2	negeqd 11386	. . . 4 ⊢ (𝜑 → -𝐷 = -𝐸)
6	1, 5	oveq12d 7386	. . 3 ⊢ (𝜑 → (𝐵 − -𝐷) = (𝐶 − -𝐸))
7	6	breq2d 5112	. 2 ⊢ (𝜑 → (𝐴 ∥ (𝐵 − -𝐷) ↔ 𝐴 ∥ (𝐶 − -𝐸)))
8	4, 7	orbi12d 919	1 ⊢ (𝜑 → ((𝐴 ∥ (𝐵 − 𝐷) ∨ 𝐴 ∥ (𝐵 − -𝐷)) ↔ (𝐴 ∥ (𝐶 − 𝐸) ∨ 𝐴 ∥ (𝐶 − -𝐸))))

Syntax hints:   → wi 4   ↔ wb 206   ∨ wo 848   = wceq 1542   class class class wbr 5100  (class class class)co 7368   − cmin 11376  -cneg 11377   ∥ cdvds 16191

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508  df-ov 7371  df-neg 11379

This theorem is referenced by:  acongrep  43334  jm2.26a  43354  jm2.26  43356