Theorem spsbc 3589

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. This is Frege's ninth axiom per Proposition 58 of [Frege1879] p. 51. See also stdpc4 2490 and rspsbc 3659. (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 2490	. . . 4 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
2		sbsbc 3580	. . . 4 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
3	1, 2	sylib 208	. . 3 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
4		dfsbcq 3578	. . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
5	3, 4	syl5ib 234	. 2 ⊢ (𝑦 = 𝐴 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑))
6	5	vtocleg 3419	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑))

Syntax hints:   → wi 4  ∀wal 1630   = wceq 1632  [wsb 2046   ∈ wcel 2139  [wsbc 3576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-12 2196  ax-13 2391  ax-ext 2740

This theorem depends on definitions:  df-bi 197  df-an 385  df-tru 1635  df-ex 1854  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-v 3342  df-sbc 3577

This theorem is referenced by:  spsbcd  3590  sbcth  3591  sbcthdv  3592  sbceqal  3628  sbcimdv  3639  sbcimdvOLD  3640  csbiebt  3694  csbexg  4944  pm14.18  39149  sbcbi  39269  onfrALTlem3  39279  csbeq2gOLD  39285  sbc3orgVD  39603  sbcbiVD  39629  csbingVD  39637  onfrALTlem3VD  39640  csbeq2gVD  39645  csbunigVD  39651