Theorem sbcalf 34404

Description: Move universal quantifier in and out of class substitution, with an explicit non-free variable condition. (Contributed by Giovanni Mascellani, 29-May-2019.)

Hypothesis

Ref	Expression
sbcalf.1	⊢ Ⅎ𝑦𝐴

Assertion

Ref	Expression
sbcalf	⊢ ([𝐴 / 𝑥]∀𝑦𝜑 ↔ ∀𝑦[𝐴 / 𝑥]𝜑)

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem sbcalf

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 2010	. . . 4 ⊢ Ⅎ𝑧𝜑
2	1	sb8 2542	. . 3 ⊢ (∀𝑦𝜑 ↔ ∀𝑧[𝑧 / 𝑦]𝜑)
3	2	sbcbii 3689	. 2 ⊢ ([𝐴 / 𝑥]∀𝑦𝜑 ↔ [𝐴 / 𝑥]∀𝑧[𝑧 / 𝑦]𝜑)
4		sbcal 3683	. 2 ⊢ ([𝐴 / 𝑥]∀𝑧[𝑧 / 𝑦]𝜑 ↔ ∀𝑧[𝐴 / 𝑥][𝑧 / 𝑦]𝜑)
5		sbcalf.1	. . . 4 ⊢ Ⅎ𝑦𝐴
6		nfs1v 2303	. . . 4 ⊢ Ⅎ𝑦[𝑧 / 𝑦]𝜑
7	5, 6	nfsbc 3655	. . 3 ⊢ Ⅎ𝑦[𝐴 / 𝑥][𝑧 / 𝑦]𝜑
8		nfv 2010	. . 3 ⊢ Ⅎ𝑧[𝐴 / 𝑥]𝜑
9		sbequ12r 2279	. . . 4 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑦]𝜑 ↔ 𝜑))
10	9	sbcbidv 3688	. . 3 ⊢ (𝑧 = 𝑦 → ([𝐴 / 𝑥][𝑧 / 𝑦]𝜑 ↔ [𝐴 / 𝑥]𝜑))
11	7, 8, 10	cbvalv1 2357	. 2 ⊢ (∀𝑧[𝐴 / 𝑥][𝑧 / 𝑦]𝜑 ↔ ∀𝑦[𝐴 / 𝑥]𝜑)
12	3, 4, 11	3bitri 289	1 ⊢ ([𝐴 / 𝑥]∀𝑦𝜑 ↔ ∀𝑦[𝐴 / 𝑥]𝜑)

Syntax hints:   ↔ wb 198  ∀wal 1651  [wsb 2064  Ⅎwnfc 2928  [wsbc 3633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-v 3387  df-sbc 3634

This theorem is referenced by:  sbcalfi  34406