Theorem spsbc 3817

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 2068 and rspsbc 3901. (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 2068	. . . 4 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
2		sbsbc 3808	. . . 4 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
3	1, 2	sylib 218	. . 3 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
4		dfsbcq 3806	. . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
5	3, 4	imbitrid 244	. 2 ⊢ (𝑦 = 𝐴 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑))
6	5	vtocleg 3565	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑))

Syntax hints:   → wi 4  ∀wal 1535   = wceq 1537  [wsb 2064   ∈ wcel 2108  [wsbc 3804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-sbc 3805

This theorem is referenced by:  spsbcd  3818  sbcth  3819  sbcthdv  3820  sbceqalOLD  3871  sbcimdvOLD  3879  csbiebt  3951  csbexg  5328  pm14.18  44397  sbcbi  44510  onfrALTlem3  44515  sbc3orgVD  44822  sbcbiVD  44847  csbingVD  44855  onfrALTlem3VD  44858  csbeq2gVD  44863  csbunigVD  44869