Theorem spsbcd 2916

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 1748 and rspsbc 2986. (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 2915	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜓 → [𝐴 / 𝑥]𝜓))
4	1, 2, 3	sylc 62	1 ⊢ (𝜑 → [𝐴 / 𝑥]𝜓)

Syntax hints:   → wi 4  ∀wal 1329   ∈ wcel 1480  [wsbc 2904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-v 2683  df-sbc 2905

This theorem is referenced by:  ovmpodxf  5889