Theorem sbi2 885

Description: Introduction of implication into substitution.

Assertion

Ref	Expression
sbi2	⊢ (([y / x]φ → [y / x]ψ) → [y / x](φ → ψ))

Proof of Theorem sbi2

Step	Hyp	Ref	Expression
1		sbn 882	. . 3 ⊢ ([y / x] ¬ φ ↔ ¬ [y / x]φ)
2		pm2.21 71	. . . 4 ⊢ (¬ φ → (φ → ψ))
3	2	sbimi 854	. . 3 ⊢ ([y / x] ¬ φ → [y / x](φ → ψ))
4	1, 3	sylbir 176	. 2 ⊢ (¬ [y / x]φ → [y / x](φ → ψ))
5		ax-1 3	. . 3 ⊢ (ψ → (φ → ψ))
6	5	sbimi 854	. 2 ⊢ ([y / x]ψ → [y / x](φ → ψ))
7	4, 6	ja 118	1 ⊢ (([y / x]φ → [y / x]ψ) → [y / x](φ → ψ))

Syntax hints:  ¬ wn 1   → wi 2  [wsb 852

This theorem is referenced by:  sbim 886

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853

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