Theorem sbcss 3660

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 ∈ B → ([̣A / x]̣C ⊆ D ↔ [A / x]C ⊆ [A / x]D))

Proof of Theorem sbcss

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcalg 3094	. . 3 ⊢ (A ∈ B → ([̣A / x]̣∀y(y ∈ C → y ∈ D) ↔ ∀y[̣A / x]̣(y ∈ C → y ∈ D)))
2		sbcimg 3087	. . . . 5 ⊢ (A ∈ B → ([̣A / x]̣(y ∈ C → y ∈ D) ↔ ([̣A / x]̣y ∈ C → [̣A / x]̣y ∈ D)))
3		sbcel2g 3157	. . . . . 6 ⊢ (A ∈ B → ([̣A / x]̣y ∈ C ↔ y ∈ [A / x]C))
4		sbcel2g 3157	. . . . . 6 ⊢ (A ∈ B → ([̣A / x]̣y ∈ D ↔ y ∈ [A / x]D))
5	3, 4	imbi12d 311	. . . . 5 ⊢ (A ∈ B → (([̣A / x]̣y ∈ C → [̣A / x]̣y ∈ D) ↔ (y ∈ [A / x]C → y ∈ [A / x]D)))
6	2, 5	bitrd 244	. . . 4 ⊢ (A ∈ B → ([̣A / x]̣(y ∈ C → y ∈ D) ↔ (y ∈ [A / x]C → y ∈ [A / x]D)))
7	6	albidv 1625	. . 3 ⊢ (A ∈ B → (∀y[̣A / x]̣(y ∈ C → y ∈ D) ↔ ∀y(y ∈ [A / x]C → y ∈ [A / x]D)))
8	1, 7	bitrd 244	. 2 ⊢ (A ∈ B → ([̣A / x]̣∀y(y ∈ C → y ∈ D) ↔ ∀y(y ∈ [A / x]C → y ∈ [A / x]D)))
9		dfss2 3262	. . 3 ⊢ (C ⊆ D ↔ ∀y(y ∈ C → y ∈ D))
10	9	sbcbii 3101	. 2 ⊢ ([̣A / x]̣C ⊆ D ↔ [̣A / x]̣∀y(y ∈ C → y ∈ D))
11		dfss2 3262	. 2 ⊢ ([A / x]C ⊆ [A / x]D ↔ ∀y(y ∈ [A / x]C → y ∈ [A / x]D))
12	8, 10, 11	3bitr4g 279	1 ⊢ (A ∈ B → ([̣A / x]̣C ⊆ D ↔ [A / x]C ⊆ [A / x]D))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   ∈ wcel 1710  [̣wsbc 3046  [csb 3136   ⊆ wss 3257

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 2478  df-v 2861  df-sbc 3047  df-csb 3137  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259

This theorem is referenced by: (None)