Theorem sbcang 3090

Description: Distribution of class substitution over conjunction. (Contributed by NM, 21-May-2004.)

Assertion

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

Proof of Theorem sbcang

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	anbi12d 691	. 2 ⊢ (y = A → (([y / x]φ ∧ [y / x]ψ) ↔ ([̣A / x]̣φ ∧ [̣A / x]̣ψ)))
5		sban 2069	. 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   ∧ wa 358   = 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:  sbcabel  3124  csbunig  3900  csbxpg  4814