Theorem spsbc 3059

Description: Specialization: if a formula is true for all sets, it is true for any class which is a set. Similar to Theorem 6.11 of [Quine] p. 44. See also stdpc4 2024 and rspsbc 3125. (Contributed by NM, 16-Jan-2004.)

Assertion

Ref	Expression
spsbc	⊢ (A ∈ V → (∀xφ → [̣A / x]̣φ))

Proof of Theorem spsbc

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		stdpc4 2024	. . . 4 ⊢ (∀xφ → [y / x]φ)
2		sbsbc 3051	. . . 4 ⊢ ([y / x]φ ↔ [̣y / x]̣φ)
3	1, 2	sylib 188	. . 3 ⊢ (∀xφ → [̣y / x]̣φ)
4		dfsbcq 3049	. . 3 ⊢ (y = A → ([̣y / x]̣φ ↔ [̣A / x]̣φ))
5	3, 4	syl5ib 210	. 2 ⊢ (y = A → (∀xφ → [̣A / x]̣φ))
6	5	vtocleg 2926	1 ⊢ (A ∈ V → (∀xφ → [̣A / x]̣φ))

Syntax hints:   → wi 4  ∀wal 1540   = 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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2862  df-sbc 3048

This theorem is referenced by:  spsbcd  3060  sbcth  3061  sbcthdv  3062  sbceqal  3098  sbcimdv  3108  csbexg  3147  csbiebt  3173