Theorem spsbc 3766

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 2069 and rspsbc 3842. (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 2069	. . . 4 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
2		sbsbc 3757	. . . 4 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
3	1, 2	sylib 218	. . 3 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
4		dfsbcq 3755	. . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
5	3, 4	imbitrid 244	. 2 ⊢ (𝑦 = 𝐴 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑))
6	5	vtocleg 3519	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑))

Syntax hints:   → wi 4  ∀wal 1538   = wceq 1540  [wsb 2065   ∈ wcel 2109  [wsbc 3753

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-sbc 3754

This theorem is referenced by:  spsbcd  3767  sbcth  3768  sbcthdv  3769  csbiebt  3891  csbexg  5265  pm14.18  44417  sbcbi  44529  onfrALTlem3  44534  sbc3orgVD  44840  sbcbiVD  44865  csbingVD  44873  onfrALTlem3VD  44876  csbeq2gVD  44881  csbunigVD  44887