Theorem sbcalfi 33919

Description: Move universal quantifier in and out of class substitution, with an explicit non-free variable condition and in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)

Hypotheses

Ref	Expression
sbcalfi.1	|- F/_ y A
sbcalfi.2	|- ( [. A / x ]. ph <-> ps )

Assertion

Ref	Expression
sbcalfi	|- ( [. A / x ]. A. y ph <-> A. y ps )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)   ps( x, y)   A( x, y)

Proof of Theorem sbcalfi

Step	Hyp	Ref	Expression
1		sbcalfi.1	. . 3 |- F/_ y A
2	1	sbcalf 33917	. 2 |- ( [. A / x ]. A. y ph <-> A. y [. A / x ]. ph )
3		sbcalfi.2	. . 3 |- ( [. A / x ]. ph <-> ps )
4	3	albii 1747	. 2 |- ( A. y [. A / x ]. ph <-> A. y ps )
5	2, 4	bitri 264	1 |- ( [. A / x ]. A. y ph <-> A. y ps )

Syntax hints:   <-> wb 196   A.wal 1481   F/_wnfc 2751   [.wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sbc 3436

This theorem is referenced by: (None)