Theorem sbi2 2393

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 2391	. . 3 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
2		pm2.21 120	. . . 4 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
3	2	sbimi 1886	. . 3 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 → [𝑦 / 𝑥](𝜑 → 𝜓))
4	1, 3	sylbir 225	. 2 ⊢ (¬ [𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥](𝜑 → 𝜓))
5		ax-1 6	. . 3 ⊢ (𝜓 → (𝜑 → 𝜓))
6	5	sbimi 1886	. 2 ⊢ ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥](𝜑 → 𝜓))
7	4, 6	ja 173	1 ⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4  [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  sbim  2395