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Theorem acongeq12d 38048
Description: Substitution deduction for alternating congruence. (Contributed by Stefan O'Rear, 3-Oct-2014.)
Hypotheses
Ref Expression
acongeq12d.1 (𝜑𝐵 = 𝐶)
acongeq12d.2 (𝜑𝐷 = 𝐸)
Assertion
Ref Expression
acongeq12d (𝜑 → ((𝐴 ∥ (𝐵𝐷) ∨ 𝐴 ∥ (𝐵 − -𝐷)) ↔ (𝐴 ∥ (𝐶𝐸) ∨ 𝐴 ∥ (𝐶 − -𝐸))))

Proof of Theorem acongeq12d
StepHypRef Expression
1 acongeq12d.1 . . . 4 (𝜑𝐵 = 𝐶)
2 acongeq12d.2 . . . 4 (𝜑𝐷 = 𝐸)
31, 2oveq12d 6831 . . 3 (𝜑 → (𝐵𝐷) = (𝐶𝐸))
43breq2d 4816 . 2 (𝜑 → (𝐴 ∥ (𝐵𝐷) ↔ 𝐴 ∥ (𝐶𝐸)))
52negeqd 10467 . . . 4 (𝜑 → -𝐷 = -𝐸)
61, 5oveq12d 6831 . . 3 (𝜑 → (𝐵 − -𝐷) = (𝐶 − -𝐸))
76breq2d 4816 . 2 (𝜑 → (𝐴 ∥ (𝐵 − -𝐷) ↔ 𝐴 ∥ (𝐶 − -𝐸)))
84, 7orbi12d 748 1 (𝜑 → ((𝐴 ∥ (𝐵𝐷) ∨ 𝐴 ∥ (𝐵 − -𝐷)) ↔ (𝐴 ∥ (𝐶𝐸) ∨ 𝐴 ∥ (𝐶 − -𝐸))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382   = wceq 1632   class class class wbr 4804  (class class class)co 6813  cmin 10458  -cneg 10459  cdvds 15182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057  df-ov 6816  df-neg 10461
This theorem is referenced by:  acongrep  38049  jm2.26a  38069  jm2.26  38071
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