Theorem sbcss 3661

Description: Distribute proper substitution through a subclass relation. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Alexander van der Vekens, 23-Jul-2017.)

Assertion

Ref	Expression
sbcss	|- (A e. B -> ( [.A / x ].C C_ D <-> [_A / x]_C C_ [_A / x]_D))

Proof of Theorem sbcss

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcalg 3095	. . 3 |- (A e. B -> ( [.A / x ].A.y(y e. C -> y e. D) <-> A.y [.A / x ].(y e. C -> y e. D)))
2		sbcimg 3088	. . . . 5 |- (A e. B -> ( [.A / x ].(y e. C -> y e. D) <-> ( [.A / x ].y e. C -> [.A / x ].y e. D)))
3		sbcel2g 3158	. . . . . 6 |- (A e. B -> ( [.A / x ].y e. C <-> y e. [_A / x]_C))
4		sbcel2g 3158	. . . . . 6 |- (A e. B -> ( [.A / x ].y e. D <-> y e. [_A / x]_D))
5	3, 4	imbi12d 311	. . . . 5 |- (A e. B -> (( [.A / x ].y e. C -> [.A / x ].y e. D) <-> (y e. [_A / x]_C -> y e. [_A / x]_D)))
6	2, 5	bitrd 244	. . . 4 |- (A e. B -> ( [.A / x ].(y e. C -> y e. D) <-> (y e. [_A / x]_C -> y e. [_A / x]_D)))
7	6	albidv 1625	. . 3 |- (A e. B -> (A.y [.A / x ].(y e. C -> y e. D) <-> A.y(y e. [_A / x]_C -> y e. [_A / x]_D)))
8	1, 7	bitrd 244	. 2 |- (A e. B -> ( [.A / x ].A.y(y e. C -> y e. D) <-> A.y(y e. [_A / x]_C -> y e. [_A / x]_D)))
9		dfss2 3263	. . 3 |- (C C_ D <-> A.y(y e. C -> y e. D))
10	9	sbcbii 3102	. 2 |- ( [.A / x ].C C_ D <-> [.A / x ].A.y(y e. C -> y e. D))
11		dfss2 3263	. 2 |- ([_A / x]_C C_ [_A / x]_D <-> A.y(y e. [_A / x]_C -> y e. [_A / x]_D))
12	8, 10, 11	3bitr4g 279	1 |- (A e. B -> ( [.A / x ].C C_ D <-> [_A / x]_C C_ [_A / x]_D))

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540   e. wcel 1710   [.wsbc 3047  [_csb 3137   C_ wss 3258

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-nan 1288  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  df-csb 3138  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260

This theorem is referenced by: (None)