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Theorem pm5.18dc 786
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 752 . 2 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2 pm5.501 237 . . . . . . . 8 (𝜑 → (¬ 𝜓 ↔ (𝜑 ↔ ¬ 𝜓)))
32a1d 22 . . . . . . 7 (𝜑 → (DECID 𝜓 → (¬ 𝜓 ↔ (𝜑 ↔ ¬ 𝜓))))
43con1biddc 779 . . . . . 6 (𝜑 → (DECID 𝜓 → (¬ (𝜑 ↔ ¬ 𝜓) ↔ 𝜓)))
54imp 119 . . . . 5 ((𝜑DECID 𝜓) → (¬ (𝜑 ↔ ¬ 𝜓) ↔ 𝜓))
6 pm5.501 237 . . . . . 6 (𝜑 → (𝜓 ↔ (𝜑𝜓)))
76adantr 265 . . . . 5 ((𝜑DECID 𝜓) → (𝜓 ↔ (𝜑𝜓)))
85, 7bitr2d 182 . . . 4 ((𝜑DECID 𝜓) → ((𝜑𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓)))
98ex 112 . . 3 (𝜑 → (DECID 𝜓 → ((𝜑𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))))
10 dcn 755 . . . . . . 7 (DECID 𝜓DECID ¬ 𝜓)
11 nbn2 621 . . . . . . . . 9 𝜑 → (¬ ¬ 𝜓 ↔ (𝜑 ↔ ¬ 𝜓)))
1211a1d 22 . . . . . . . 8 𝜑 → (DECID ¬ 𝜓 → (¬ ¬ 𝜓 ↔ (𝜑 ↔ ¬ 𝜓))))
1312con1biddc 779 . . . . . . 7 𝜑 → (DECID ¬ 𝜓 → (¬ (𝜑 ↔ ¬ 𝜓) ↔ ¬ 𝜓)))
1410, 13syl5 32 . . . . . 6 𝜑 → (DECID 𝜓 → (¬ (𝜑 ↔ ¬ 𝜓) ↔ ¬ 𝜓)))
1514imp 119 . . . . 5 ((¬ 𝜑DECID 𝜓) → (¬ (𝜑 ↔ ¬ 𝜓) ↔ ¬ 𝜓))
16 nbn2 621 . . . . . 6 𝜑 → (¬ 𝜓 ↔ (𝜑𝜓)))
1716adantr 265 . . . . 5 ((¬ 𝜑DECID 𝜓) → (¬ 𝜓 ↔ (𝜑𝜓)))
1815, 17bitr2d 182 . . . 4 ((¬ 𝜑DECID 𝜓) → ((𝜑𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓)))
1918ex 112 . . 3 𝜑 → (DECID 𝜓 → ((𝜑𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))))
209, 19jaoi 644 . 2 ((𝜑 ∨ ¬ 𝜑) → (DECID 𝜓 → ((𝜑𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))))
211, 20sylbi 118 1 (DECID 𝜑 → (DECID 𝜓 → ((𝜑𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wb 102  wo 637  DECID wdc 751
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 552  ax-in2 553  ax-io 638
This theorem depends on definitions:  df-bi 114  df-dc 752
This theorem is referenced by:  xor3dc  1292  dfbi3dc  1302
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