Theorem acongeq12d 42301

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 7423	. . 3 ⊢ (𝜑 → (𝐵 − 𝐷) = (𝐶 − 𝐸))
4	3	breq2d 5153	. 2 ⊢ (𝜑 → (𝐴 ∥ (𝐵 − 𝐷) ↔ 𝐴 ∥ (𝐶 − 𝐸)))
5	2	negeqd 11458	. . . 4 ⊢ (𝜑 → -𝐷 = -𝐸)
6	1, 5	oveq12d 7423	. . 3 ⊢ (𝜑 → (𝐵 − -𝐷) = (𝐶 − -𝐸))
7	6	breq2d 5153	. 2 ⊢ (𝜑 → (𝐴 ∥ (𝐵 − -𝐷) ↔ 𝐴 ∥ (𝐶 − -𝐸)))
8	4, 7	orbi12d 915	1 ⊢ (𝜑 → ((𝐴 ∥ (𝐵 − 𝐷) ∨ 𝐴 ∥ (𝐵 − -𝐷)) ↔ (𝐴 ∥ (𝐶 − 𝐸) ∨ 𝐴 ∥ (𝐶 − -𝐸))))

Syntax hints:   → wi 4   ↔ wb 205   ∨ wo 844   = wceq 1533   class class class wbr 5141  (class class class)co 7405   − cmin 11448  -cneg 11449   ∥ cdvds 16204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6489  df-fv 6545  df-ov 7408  df-neg 11451

This theorem is referenced by:  acongrep  42302  jm2.26a  42322  jm2.26  42324