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Theorem acongeq12d 41346
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 7376 . . 3 (𝜑 → (𝐵𝐷) = (𝐶𝐸))
43breq2d 5118 . 2 (𝜑 → (𝐴 ∥ (𝐵𝐷) ↔ 𝐴 ∥ (𝐶𝐸)))
52negeqd 11400 . . . 4 (𝜑 → -𝐷 = -𝐸)
61, 5oveq12d 7376 . . 3 (𝜑 → (𝐵 − -𝐷) = (𝐶 − -𝐸))
76breq2d 5118 . 2 (𝜑 → (𝐴 ∥ (𝐵 − -𝐷) ↔ 𝐴 ∥ (𝐶 − -𝐸)))
84, 7orbi12d 918 1 (𝜑 → ((𝐴 ∥ (𝐵𝐷) ∨ 𝐴 ∥ (𝐵 − -𝐷)) ↔ (𝐴 ∥ (𝐶𝐸) ∨ 𝐴 ∥ (𝐶 − -𝐸))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wo 846   = wceq 1542   class class class wbr 5106  (class class class)co 7358  cmin 11390  -cneg 11391  cdvds 16141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-ov 7361  df-neg 11393
This theorem is referenced by:  acongrep  41347  jm2.26a  41367  jm2.26  41369
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