Theorem spsbc 3785

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 2073 and rspsbc 3862. (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 2073	. . . 4 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
2		sbsbc 3776	. . . 4 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
3	1, 2	sylib 220	. . 3 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
4		dfsbcq 3774	. . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
5	3, 4	syl5ib 246	. 2 ⊢ (𝑦 = 𝐴 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑))
6	5	vtocleg 3581	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑))

Syntax hints:   → wi 4  ∀wal 1535   = wceq 1537  [wsb 2069   ∈ wcel 2114  [wsbc 3772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-sbc 3773

This theorem is referenced by:  spsbcd  3786  sbcth  3787  sbcthdv  3788  sbceqal  3835  sbcimdv  3843  csbiebt  3912  csbexg  5214  pm14.18  40780  sbcbi  40893  onfrALTlem3  40898  sbc3orgVD  41205  sbcbiVD  41230  csbingVD  41238  onfrALTlem3VD  41241  csbeq2gVD  41246  csbunigVD  41252