Theorem sbi2 2335

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 2313	. . 3 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
2		pm2.21 124	. . . 4 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
3	2	sbimi 2110	. . 3 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 → [𝑦 / 𝑥](𝜑 → 𝜓))
4	1, 3	sylbir 238	. 2 ⊢ (¬ [𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥](𝜑 → 𝜓))
5		ax-1 6	. . 3 ⊢ (𝜓 → (𝜑 → 𝜓))
6	5	sbimi 2110	. 2 ⊢ ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥](𝜑 → 𝜓))
7	4, 6	ja 188	1 ⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑 → 𝜓))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  [wsb 2098

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-ex 1812  df-nf 1816  df-sb 2099

This theorem is used by:  sbim  2336