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Theorem acongeq12d 36564
 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 6567 . . 3 (𝜑 → (𝐵𝐷) = (𝐶𝐸))
43breq2d 4595 . 2 (𝜑 → (𝐴 ∥ (𝐵𝐷) ↔ 𝐴 ∥ (𝐶𝐸)))
52negeqd 10154 . . . 4 (𝜑 → -𝐷 = -𝐸)
61, 5oveq12d 6567 . . 3 (𝜑 → (𝐵 − -𝐷) = (𝐶 − -𝐸))
76breq2d 4595 . 2 (𝜑 → (𝐴 ∥ (𝐵 − -𝐷) ↔ 𝐴 ∥ (𝐶 − -𝐸)))
84, 7orbi12d 742 1 (𝜑 → ((𝐴 ∥ (𝐵𝐷) ∨ 𝐴 ∥ (𝐵 − -𝐷)) ↔ (𝐴 ∥ (𝐶𝐸) ∨ 𝐴 ∥ (𝐶 − -𝐸))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   = wceq 1475   class class class wbr 4583  (class class class)co 6549   − cmin 10145  -cneg 10146   ∥ cdvds 14821 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-neg 10148 This theorem is referenced by:  acongrep  36565  jm2.26a  36585  jm2.26  36587
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