Theorem a4sbc 1444

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.

Assertion

Ref	Expression
a4sbc	⊢ (A ∈ B → (∀xφ → [A / x]φ))

Proof of Theorem a4sbc

Step	Hyp	Ref	Expression
1		dfsbcq 1442	. . 3 ⊢ (y = A → ([y / x]φ ↔ [A / x]φ))
2		stdpc4 869	. . 3 ⊢ (∀xφ → [y / x]φ)
3	1, 2	syl5bi 183	. 2 ⊢ (y = A → (∀xφ → [A / x]φ))
4	3	vtocleg 1390	1 ⊢ (A ∈ B → (∀xφ → [A / x]φ))

Syntax hints:   → wi 2  ∀wal 672  [wsb 852   = wceq 1091   ∈ wcel 1092  [wsbc 1440

This theorem is referenced by:  bisbcdv 1468  rax4 1471

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-9 799  ax-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-sbc 1441

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