Theorem spsbcd 3767

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 2105 and rspsbc 3841. (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 3766	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜓 → [𝐴 / 𝑥]𝜓))
4	1, 2, 3	sylc 66	1 ⊢ (𝜑 → [𝐴 / 𝑥]𝜓)

Syntax hints:   → wi 4  ∀wal 1565   ∈ wcel 2149  [wsbc 3753

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-sbc 3754

This theorem is referenced by:  ovmpodxf  7561  ex-natded9.26  30710  spsbcdi  38656  ovmpordxf  49003