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Theorem axorbciffatcxorb 41715
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 41713 . . . 4 ¬ (𝜑𝜓)
3 xor3 373 . . . 4 (¬ (𝜑𝜓) ↔ (𝜑 ↔ ¬ 𝜓))
42, 3mpbi 221 . . 3 (𝜑 ↔ ¬ 𝜓)
5 axorbciffatcxorb.2 . . 3 (𝜒𝜑)
64, 5aiffnbandciffatnotciffb 41714 . 2 ¬ (𝜒𝜓)
7 df-xor 1634 . 2 ((𝜒𝜓) ↔ ¬ (𝜒𝜓))
86, 7mpbir 222 1 (𝜒𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 197  wxo 1633
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 198  df-xor 1634
This theorem is referenced by:  mdandyvrx0  41791  mdandyvrx1  41792  mdandyvrx2  41793  mdandyvrx3  41794  mdandyvrx4  41795  mdandyvrx5  41796  mdandyvrx6  41797  mdandyvrx7  41798
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