Mathbox for Jarvin Udandy < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  axorbciffatcxorb Structured version   Visualization version   GIF version

Theorem axorbciffatcxorb 41393
 Description: Given a is equivalent to (not b), c is equivalent to a. there exists a proof for ( c xor b ) . (Contributed by Jarvin Udandy, 7-Sep-2016.)
Hypotheses
Ref Expression
axorbciffatcxorb.1 (𝜑𝜓)
axorbciffatcxorb.2 (𝜒𝜑)
Assertion
Ref Expression
axorbciffatcxorb (𝜒𝜓)

Proof of Theorem axorbciffatcxorb
StepHypRef Expression
1 axorbciffatcxorb.1 . . . . 5 (𝜑𝜓)
21axorbtnotaiffb 41391 . . . 4 ¬ (𝜑𝜓)
3 xor3 371 . . . 4 (¬ (𝜑𝜓) ↔ (𝜑 ↔ ¬ 𝜓))
42, 3mpbi 220 . . 3 (𝜑 ↔ ¬ 𝜓)
5 axorbciffatcxorb.2 . . 3 (𝜒𝜑)
64, 5aiffnbandciffatnotciffb 41392 . 2 ¬ (𝜒𝜓)
7 df-xor 1505 . 2 ((𝜒𝜓) ↔ ¬ (𝜒𝜓))
86, 7mpbir 221 1 (𝜒𝜓)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 196   ⊻ wxo 1504 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8 This theorem depends on definitions:  df-bi 197  df-xor 1505 This theorem is referenced by:  mdandyvrx0  41469  mdandyvrx1  41470  mdandyvrx2  41471  mdandyvrx3  41472  mdandyvrx4  41473  mdandyvrx5  41474  mdandyvrx6  41475  mdandyvrx7  41476
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