Theorem spsbcd 3060

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 Mario Carneiro, 9-Feb-2017.)

Hypotheses

Ref	Expression
spsbcd.1	⊢ (φ → A ∈ V)
spsbcd.2	⊢ (φ → ∀xψ)

Assertion

Ref	Expression
spsbcd	⊢ (φ → [̣A / x]̣ψ)

Proof of Theorem spsbcd

Step	Hyp	Ref	Expression
1		spsbcd.1	. 2 ⊢ (φ → A ∈ V)
2		spsbcd.2	. 2 ⊢ (φ → ∀xψ)
3		spsbc 3059	. 2 ⊢ (A ∈ V → (∀xψ → [̣A / x]̣ψ))
4	1, 2, 3	sylc 56	1 ⊢ (φ → [̣A / x]̣ψ)

Syntax hints:   → wi 4  ∀wal 1540   ∈ 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: (None)