Theorem sbc19.21g 3111

Description: Substitution for a variable not free in antecedent affects only the consequent. (Contributed by NM, 11-Oct-2004.)

Hypothesis

Ref	Expression
sbcgf.1	⊢ Ⅎxφ

Assertion

Ref	Expression
sbc19.21g	⊢ (A ∈ V → ([̣A / x]̣(φ → ψ) ↔ (φ → [̣A / x]̣ψ)))

Proof of Theorem sbc19.21g

Step	Hyp	Ref	Expression
1		sbcimg 3088	. 2 ⊢ (A ∈ V → ([̣A / x]̣(φ → ψ) ↔ ([̣A / x]̣φ → [̣A / x]̣ψ)))
2		sbcgf.1	. . . 4 ⊢ Ⅎxφ
3	2	sbcgf 3110	. . 3 ⊢ (A ∈ V → ([̣A / x]̣φ ↔ φ))
4	3	imbi1d 308	. 2 ⊢ (A ∈ V → (([̣A / x]̣φ → [̣A / x]̣ψ) ↔ (φ → [̣A / x]̣ψ)))
5	1, 4	bitrd 244	1 ⊢ (A ∈ V → ([̣A / x]̣(φ → ψ) ↔ (φ → [̣A / x]̣ψ)))

Syntax hints:   → wi 4   ↔ wb 176  Ⅎwnf 1544   ∈ wcel 1710  [̣wsbc 3047

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  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048

This theorem is referenced by: (None)