Theorem sbi2 2301

Description: Introduction of implication into substitution. (Contributed by NM, 14-May-1993.)

Assertion

Ref	Expression
sbi2	⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑 → 𝜓))

Proof of Theorem sbi2

Step	Hyp	Ref	Expression
1		sbn 2279	. . 3 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
2		pm2.21 123	. . . 4 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
3	2	sbimi 2073	. . 3 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 → [𝑦 / 𝑥](𝜑 → 𝜓))
4	1, 3	sylbir 235	. 2 ⊢ (¬ [𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥](𝜑 → 𝜓))
5		ax-1 6	. . 3 ⊢ (𝜓 → (𝜑 → 𝜓))
6	5	sbimi 2073	. 2 ⊢ ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥](𝜑 → 𝜓))
7	4, 6	ja 186	1 ⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4  [wsb 2063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-10 2140  ax-12 2176

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-nf 1783  df-sb 2064

This theorem is referenced by:  sbim  2302