Theorem sbi2 2313

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 2291	. . 3 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
2		pm2.21 123	. . . 4 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
3	2	sbimi 2085	. . 3 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 → [𝑦 / 𝑥](𝜑 → 𝜓))
4	1, 3	sylbir 236	. 2 ⊢ (¬ [𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥](𝜑 → 𝜓))
5		ax-1 6	. . 3 ⊢ (𝜓 → (𝜑 → 𝜓))
6	5	sbimi 2085	. 2 ⊢ ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥](𝜑 → 𝜓))
7	4, 6	ja 187	1 ⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4  [wsb 2073

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-10 2152  ax-12 2189

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-nf 1791  df-sb 2074

This theorem is referenced by:  sbim  2314