Theorem sbrim 2067

Description: Substitution with a variable not free in antecedent affects only the consequent. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)

Hypothesis

Ref	Expression
sbrim.1	⊢ Ⅎxφ

Assertion

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

Proof of Theorem sbrim

Step	Hyp	Ref	Expression
1		sbim 2065	. 2 ⊢ ([y / x](φ → ψ) ↔ ([y / x]φ → [y / x]ψ))
2		sbrim.1	. . . 4 ⊢ Ⅎxφ
3	2	sbf 2026	. . 3 ⊢ ([y / x]φ ↔ φ)
4	3	imbi1i 315	. 2 ⊢ (([y / x]φ → [y / x]ψ) ↔ (φ → [y / x]ψ))
5	1, 4	bitri 240	1 ⊢ ([y / x](φ → ψ) ↔ (φ → [y / x]ψ))

Syntax hints:   → wi 4   ↔ wb 176  Ⅎwnf 1544  [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:  sbco2d  2087  2mos  2283