Theorem sbi2 2469

Description: Introduction of implication into substitution. For a version requiring disjoint variables, but fewer axioms, see sbi2v 2279. (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 2467	. . 3 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
2		pm2.21 121	. . . 4 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
3	2	sbimi 2017	. . 3 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 → [𝑦 / 𝑥](𝜑 → 𝜓))
4	1, 3	sylbir 227	. 2 ⊢ (¬ [𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥](𝜑 → 𝜓))
5		ax-1 6	. . 3 ⊢ (𝜓 → (𝜑 → 𝜓))
6	5	sbimi 2017	. 2 ⊢ ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥](𝜑 → 𝜓))
7	4, 6	ja 175	1 ⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4  [wsb 2011

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-10 2135  ax-12 2163  ax-13 2334

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-ex 1824  df-nf 1828  df-sb 2012

This theorem is referenced by:  sbim  2471