Theorem spsbcd 3730

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 2071 and rspsbc 3812. (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 3729	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜓 → [𝐴 / 𝑥]𝜓))
4	1, 2, 3	sylc 65	1 ⊢ (𝜑 → [𝐴 / 𝑥]𝜓)

Syntax hints:   → wi 4  ∀wal 1537   ∈ wcel 2106  [wsbc 3716

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-sbc 3717

This theorem is referenced by:  ovmpodxf  7423  ex-natded9.26  28783  spsbcdi  36276  ovmpordxf  45674