Theorem ichim 46934

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 46933	. . . 4 ⊢ ([𝑎⇄𝑏]𝜓 ↔ [𝑎⇄𝑏] ¬ 𝜓)
2		ichan 46932	. . . 4 ⊢ (([𝑎⇄𝑏]𝜑 ∧ [𝑎⇄𝑏] ¬ 𝜓) → [𝑎⇄𝑏](𝜑 ∧ ¬ 𝜓))
3	1, 2	sylan2b 592	. . 3 ⊢ (([𝑎⇄𝑏]𝜑 ∧ [𝑎⇄𝑏]𝜓) → [𝑎⇄𝑏](𝜑 ∧ ¬ 𝜓))
4		ichn 46933	. . 3 ⊢ ([𝑎⇄𝑏](𝜑 ∧ ¬ 𝜓) ↔ [𝑎⇄𝑏] ¬ (𝜑 ∧ ¬ 𝜓))
5	3, 4	sylib 217	. 2 ⊢ (([𝑎⇄𝑏]𝜑 ∧ [𝑎⇄𝑏]𝜓) → [𝑎⇄𝑏] ¬ (𝜑 ∧ ¬ 𝜓))
6		iman 400	. . . . 5 ⊢ ((𝜑 → 𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓))
7	6	a1i 11	. . . 4 ⊢ (⊤ → ((𝜑 → 𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓)))
8	7	ichbidv 46930	. . 3 ⊢ (⊤ → ([𝑎⇄𝑏](𝜑 → 𝜓) ↔ [𝑎⇄𝑏] ¬ (𝜑 ∧ ¬ 𝜓)))
9	8	mptru 1540	. 2 ⊢ ([𝑎⇄𝑏](𝜑 → 𝜓) ↔ [𝑎⇄𝑏] ¬ (𝜑 ∧ ¬ 𝜓))
10	5, 9	sylibr 233	1 ⊢ (([𝑎⇄𝑏]𝜑 ∧ [𝑎⇄𝑏]𝜓) → [𝑎⇄𝑏](𝜑 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394  ⊤wtru 1534  [wich 46922

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-12 2166

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-ich 46923

This theorem is referenced by: (None)