Theorem sbcal 3094

Description: Move universal quantifier in and out of class substitution. (Contributed by NM, 31-Dec-2016.)

Assertion

Ref	Expression
sbcal	⊢ ([̣A / y]̣∀xφ ↔ ∀x[̣A / y]̣φ)

Distinct variable groups:   x,A   x,y

Allowed substitution hints:   φ(x,y)   A(y)

Proof of Theorem sbcal

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcex 3056	. 2 ⊢ ([̣A / y]̣∀xφ → A ∈ V)
2		sbcex 3056	. . 3 ⊢ ([̣A / y]̣φ → A ∈ V)
3	2	sps 1754	. 2 ⊢ (∀x[̣A / y]̣φ → A ∈ V)
4		dfsbcq2 3050	. . 3 ⊢ (z = A → ([z / y]∀xφ ↔ [̣A / y]̣∀xφ))
5		dfsbcq2 3050	. . . 4 ⊢ (z = A → ([z / y]φ ↔ [̣A / y]̣φ))
6	5	albidv 1625	. . 3 ⊢ (z = A → (∀x[z / y]φ ↔ ∀x[̣A / y]̣φ))
7		sbal 2127	. . 3 ⊢ ([z / y]∀xφ ↔ ∀x[z / y]φ)
8	4, 6, 7	vtoclbg 2916	. 2 ⊢ (A ∈ V → ([̣A / y]̣∀xφ ↔ ∀x[̣A / y]̣φ))
9	1, 3, 8	pm5.21nii 342	1 ⊢ ([̣A / y]̣∀xφ ↔ ∀x[̣A / y]̣φ)

Syntax hints:   ↔ wb 176  ∀wal 1540   = wceq 1642  [wsb 1648   ∈ wcel 1710  Vcvv 2860  [̣wsbc 3047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048

This theorem is referenced by: (None)