Theorem sbcimg 3088

Description: Distribution of class substitution over implication. (Contributed by NM, 16-Jan-2004.)

Assertion

Ref	Expression
sbcimg	⊢ (A ∈ V → ([̣A / x]̣(φ → ψ) ↔ ([̣A / x]̣φ → [̣A / x]̣ψ)))

Proof of Theorem sbcimg

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 3050	. 2 ⊢ (y = A → ([y / x](φ → ψ) ↔ [̣A / x]̣(φ → ψ)))
2		dfsbcq2 3050	. . 3 ⊢ (y = A → ([y / x]φ ↔ [̣A / x]̣φ))
3		dfsbcq2 3050	. . 3 ⊢ (y = A → ([y / x]ψ ↔ [̣A / x]̣ψ))
4	2, 3	imbi12d 311	. 2 ⊢ (y = A → (([y / x]φ → [y / x]ψ) ↔ ([̣A / x]̣φ → [̣A / x]̣ψ)))
5		sbim 2065	. 2 ⊢ ([y / x](φ → ψ) ↔ ([y / x]φ → [y / x]ψ))
6	1, 4, 5	vtoclbg 2916	1 ⊢ (A ∈ V → ([̣A / x]̣(φ → ψ) ↔ ([̣A / x]̣φ → [̣A / x]̣ψ)))

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642  [wsb 1648   ∈ 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:  sbceqal  3098  sbcimdv  3108  sbc19.21g  3111  sbcss  3661  iota4an  4359