Theorem spsbc 2966

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 1768 and rspsbc 3037. (Contributed by NM, 16-Jan-2004.)

Assertion

Ref	Expression
spsbc	⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑))

Proof of Theorem spsbc

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		stdpc4 1768	. . . 4 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
2		sbsbc 2959	. . . 4 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
3	1, 2	sylib 121	. . 3 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
4		dfsbcq 2957	. . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
5	3, 4	syl5ib 153	. 2 ⊢ (𝑦 = 𝐴 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑))
6	5	vtocleg 2801	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑))

Syntax hints:   → wi 4  ∀wal 1346   = wceq 1348  [wsb 1755   ∈ wcel 2141  [wsbc 2955

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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732  df-sbc 2956

This theorem is referenced by:  spsbcd  2967  sbcth  2968  sbcthdv  2969  sbceqal  3010  sbcimdv  3020  csbiebt  3088  csbexga  4117