Theorem spsbcd 2852

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 1705 and rspsbc 2921. (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 2851	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜓 → [𝐴 / 𝑥]𝜓))
4	1, 2, 3	sylc 61	1 ⊢ (𝜑 → [𝐴 / 𝑥]𝜓)

Syntax hints:   → wi 4  ∀wal 1287   ∈ wcel 1438  [wsbc 2840

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621  df-sbc 2841

This theorem is referenced by:  ovmpt2dxf  5770