Theorem ichim 44327

Description: Formula building rule for implication in interchangeability. (Contributed by SN, 4-May-2024.)

Assertion

Ref	Expression
ichim	⊢ (([𝑎⇄𝑏]𝜑 ∧ [𝑎⇄𝑏]𝜓) → [𝑎⇄𝑏](𝜑 → 𝜓))

Proof of Theorem ichim

Step	Hyp	Ref	Expression
1		ichn 44326	. . . 4 ⊢ ([𝑎⇄𝑏]𝜓 ↔ [𝑎⇄𝑏] ¬ 𝜓)
2		ichan 44325	. . . 4 ⊢ (([𝑎⇄𝑏]𝜑 ∧ [𝑎⇄𝑏] ¬ 𝜓) → [𝑎⇄𝑏](𝜑 ∧ ¬ 𝜓))
3	1, 2	sylan2b 597	. . 3 ⊢ (([𝑎⇄𝑏]𝜑 ∧ [𝑎⇄𝑏]𝜓) → [𝑎⇄𝑏](𝜑 ∧ ¬ 𝜓))
4		ichn 44326	. . 3 ⊢ ([𝑎⇄𝑏](𝜑 ∧ ¬ 𝜓) ↔ [𝑎⇄𝑏] ¬ (𝜑 ∧ ¬ 𝜓))
5	3, 4	sylib 221	. 2 ⊢ (([𝑎⇄𝑏]𝜑 ∧ [𝑎⇄𝑏]𝜓) → [𝑎⇄𝑏] ¬ (𝜑 ∧ ¬ 𝜓))
6		iman 406	. . . . 5 ⊢ ((𝜑 → 𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓))
7	6	a1i 11	. . . 4 ⊢ (⊤ → ((𝜑 → 𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓)))
8	7	ichbidv 44323	. . 3 ⊢ (⊤ → ([𝑎⇄𝑏](𝜑 → 𝜓) ↔ [𝑎⇄𝑏] ¬ (𝜑 ∧ ¬ 𝜓)))
9	8	mptru 1546	. 2 ⊢ ([𝑎⇄𝑏](𝜑 → 𝜓) ↔ [𝑎⇄𝑏] ¬ (𝜑 ∧ ¬ 𝜓))
10	5, 9	sylibr 237	1 ⊢ (([𝑎⇄𝑏]𝜑 ∧ [𝑎⇄𝑏]𝜓) → [𝑎⇄𝑏](𝜑 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400  ⊤wtru 1540  [wich 44315

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-10 2143  ax-12 2176

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-ich 44316

This theorem is referenced by: (None)