Theorem ichim 48633

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 48632	. . . 4 ⊢ ([𝑎⇄𝑏]𝜓 ↔ [𝑎⇄𝑏] ¬ 𝜓)
2		ichan 48631	. . . 4 ⊢ (([𝑎⇄𝑏]𝜑 ∧ [𝑎⇄𝑏] ¬ 𝜓) → [𝑎⇄𝑏](𝜑 ∧ ¬ 𝜓))
3	1, 2	sylan2b 606	. . 3 ⊢ (([𝑎⇄𝑏]𝜑 ∧ [𝑎⇄𝑏]𝜓) → [𝑎⇄𝑏](𝜑 ∧ ¬ 𝜓))
4		ichn 48632	. . 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 407	. . . . 5 ⊢ ((𝜑 → 𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓))
7	6	a1i 11	. . . 4 ⊢ (⊤ → ((𝜑 → 𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓)))
8	7	ichbidv 48629	. . 3 ⊢ (⊤ → ([𝑎⇄𝑏](𝜑 → 𝜓) ↔ [𝑎⇄𝑏] ¬ (𝜑 ∧ ¬ 𝜓)))
9	8	mptru 1576	. 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))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ⊤wtru 1570  [wich 48621

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2099  df-ich 48622

This theorem is used by: (None)