Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  acongeq12d Structured version   Visualization version   GIF version

Theorem acongeq12d 42465
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 7434 . . 3 (𝜑 → (𝐵𝐷) = (𝐶𝐸))
43breq2d 5155 . 2 (𝜑 → (𝐴 ∥ (𝐵𝐷) ↔ 𝐴 ∥ (𝐶𝐸)))
52negeqd 11484 . . . 4 (𝜑 → -𝐷 = -𝐸)
61, 5oveq12d 7434 . . 3 (𝜑 → (𝐵 − -𝐷) = (𝐶 − -𝐸))
76breq2d 5155 . 2 (𝜑 → (𝐴 ∥ (𝐵 − -𝐷) ↔ 𝐴 ∥ (𝐶 − -𝐸)))
84, 7orbi12d 916 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 845   = wceq 1533   class class class wbr 5143  (class class class)co 7416  cmin 11474  -cneg 11475  cdvds 16230
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 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-iota 6495  df-fv 6551  df-ov 7419  df-neg 11477
This theorem is referenced by:  acongrep  42466  jm2.26a  42486  jm2.26  42488
  Copyright terms: Public domain W3C validator