Theorem sbi2 2064

Description: Introduction of implication into substitution. (Contributed by NM, 5-Aug-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 2062	. . 3 ⊢ ([y / x] ¬ φ ↔ ¬ [y / x]φ)
2		pm2.21 100	. . . 4 ⊢ (¬ φ → (φ → ψ))
3	2	sbimi 1652	. . 3 ⊢ ([y / x] ¬ φ → [y / x](φ → ψ))
4	1, 3	sylbir 204	. 2 ⊢ (¬ [y / x]φ → [y / x](φ → ψ))
5		ax-1 6	. . 3 ⊢ (ψ → (φ → ψ))
6	5	sbimi 1652	. 2 ⊢ ([y / x]ψ → [y / x](φ → ψ))
7	4, 6	ja 153	1 ⊢ (([y / x]φ → [y / x]ψ) → [y / x](φ → ψ))

Syntax hints:  ¬ wn 3   → wi 4  [wsb 1648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is referenced by:  sbim  2065