Theorem sbcalf 37581

Description: Move universal quantifier in and out of class substitution, with an explicit nonfree 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		sb8v 2344	. . 3 ⊢ (∀𝑦𝜑 ↔ ∀𝑧[𝑧 / 𝑦]𝜑)
2	1	sbcbii 3835	. 2 ⊢ ([𝐴 / 𝑥]∀𝑦𝜑 ↔ [𝐴 / 𝑥]∀𝑧[𝑧 / 𝑦]𝜑)
3		sbcal 3838	. 2 ⊢ ([𝐴 / 𝑥]∀𝑧[𝑧 / 𝑦]𝜑 ↔ ∀𝑧[𝐴 / 𝑥][𝑧 / 𝑦]𝜑)
4		sbcalf.1	. . . 4 ⊢ Ⅎ𝑦𝐴
5		nfs1v 2146	. . . 4 ⊢ Ⅎ𝑦[𝑧 / 𝑦]𝜑
6	4, 5	nfsbcw 3797	. . 3 ⊢ Ⅎ𝑦[𝐴 / 𝑥][𝑧 / 𝑦]𝜑
7		nfv 1910	. . 3 ⊢ Ⅎ𝑧[𝐴 / 𝑥]𝜑
8		sbequ12r 2240	. . . 4 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑦]𝜑 ↔ 𝜑))
9	8	sbcbidv 3834	. . 3 ⊢ (𝑧 = 𝑦 → ([𝐴 / 𝑥][𝑧 / 𝑦]𝜑 ↔ [𝐴 / 𝑥]𝜑))
10	6, 7, 9	cbvalv1 2333	. 2 ⊢ (∀𝑧[𝐴 / 𝑥][𝑧 / 𝑦]𝜑 ↔ ∀𝑦[𝐴 / 𝑥]𝜑)
11	2, 3, 10	3bitri 297	1 ⊢ ([𝐴 / 𝑥]∀𝑦𝜑 ↔ ∀𝑦[𝐴 / 𝑥]𝜑)

Syntax hints:   ↔ wb 205  ∀wal 1532  [wsb 2060  Ⅎwnfc 2879  [wsbc 3775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-v 3472  df-sbc 3776

This theorem is referenced by:  sbcalfi  37583