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Theorem pm5.18dc 776
Description: Relationship between an equivalence and an equivalence with some negation, for decidable propositions. Based on theorem *5.18 of [WhiteheadRussell] p. 124. Given decidability, we can consider ¬ (φ ↔ ¬ ψ) to represent "negated exclusive-or". (Contributed by Jim Kingdon, 4-Apr-2018.)
Assertion
Ref Expression
pm5.18dc (DECID φ → (DECID ψ → ((φψ) ↔ ¬ (φ ↔ ¬ ψ))))

Proof of Theorem pm5.18dc
StepHypRef Expression
1 df-dc 742 . 2 (DECID φ ↔ (φ ¬ φ))
2 pm5.501 233 . . . . . . . 8 (φ → (¬ ψ ↔ (φ ↔ ¬ ψ)))
32a1d 22 . . . . . . 7 (φ → (DECID ψ → (¬ ψ ↔ (φ ↔ ¬ ψ))))
43con1biddc 769 . . . . . 6 (φ → (DECID ψ → (¬ (φ ↔ ¬ ψ) ↔ ψ)))
54imp 115 . . . . 5 ((φ DECID ψ) → (¬ (φ ↔ ¬ ψ) ↔ ψ))
6 pm5.501 233 . . . . . 6 (φ → (ψ ↔ (φψ)))
76adantr 261 . . . . 5 ((φ DECID ψ) → (ψ ↔ (φψ)))
85, 7bitr2d 178 . . . 4 ((φ DECID ψ) → ((φψ) ↔ ¬ (φ ↔ ¬ ψ)))
98ex 108 . . 3 (φ → (DECID ψ → ((φψ) ↔ ¬ (φ ↔ ¬ ψ))))
10 dcn 745 . . . . . . 7 (DECID ψDECID ¬ ψ)
11 nbn2 612 . . . . . . . . 9 φ → (¬ ¬ ψ ↔ (φ ↔ ¬ ψ)))
1211a1d 22 . . . . . . . 8 φ → (DECID ¬ ψ → (¬ ¬ ψ ↔ (φ ↔ ¬ ψ))))
1312con1biddc 769 . . . . . . 7 φ → (DECID ¬ ψ → (¬ (φ ↔ ¬ ψ) ↔ ¬ ψ)))
1410, 13syl5 28 . . . . . 6 φ → (DECID ψ → (¬ (φ ↔ ¬ ψ) ↔ ¬ ψ)))
1514imp 115 . . . . 5 ((¬ φ DECID ψ) → (¬ (φ ↔ ¬ ψ) ↔ ¬ ψ))
16 nbn2 612 . . . . . 6 φ → (¬ ψ ↔ (φψ)))
1716adantr 261 . . . . 5 ((¬ φ DECID ψ) → (¬ ψ ↔ (φψ)))
1815, 17bitr2d 178 . . . 4 ((¬ φ DECID ψ) → ((φψ) ↔ ¬ (φ ↔ ¬ ψ)))
1918ex 108 . . 3 φ → (DECID ψ → ((φψ) ↔ ¬ (φ ↔ ¬ ψ))))
209, 19jaoi 635 . 2 ((φ ¬ φ) → (DECID ψ → ((φψ) ↔ ¬ (φ ↔ ¬ ψ))))
211, 20sylbi 114 1 (DECID φ → (DECID ψ → ((φψ) ↔ ¬ (φ ↔ ¬ ψ))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wo 628  DECID wdc 741
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629
This theorem depends on definitions:  df-bi 110  df-dc 742
This theorem is referenced by:  xor3dc  1275  dfbi3dc  1285
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