Theorem nbbndc 1285

Description: Move negation outside of biconditional, for decidable propositions. Compare Theorem *5.18 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 18-Apr-2018.)

Assertion

Ref	Expression
nbbndc	⊢ (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))))

Proof of Theorem nbbndc

Step	Hyp	Ref	Expression
1		xor3dc 1278	. . . . 5 ⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓))))
2	1	imp 115	. . . 4 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓)))
3		con2bidc 769	. . . . 5 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ↔ ¬ 𝜓) ↔ (𝜓 ↔ ¬ 𝜑))))
4	3	imp 115	. . . 4 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((𝜑 ↔ ¬ 𝜓) ↔ (𝜓 ↔ ¬ 𝜑)))
5	2, 4	bitrd 177	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (¬ (𝜑 ↔ 𝜓) ↔ (𝜓 ↔ ¬ 𝜑)))
6		bicom 128	. . 3 ⊢ ((𝜓 ↔ ¬ 𝜑) ↔ (¬ 𝜑 ↔ 𝜓))
7	5, 6	syl6rbb 186	. 2 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((¬ 𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓)))
8	7	ex 108	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98  DECID wdc 742

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 630

This theorem depends on definitions:  df-bi 110  df-dc 743

This theorem is referenced by:  biassdc  1286