Theorem sbcssg 4483

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
sbcssg	⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 ⊆ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ⊆ ⦋𝐴 / 𝑥⦌𝐶))

Proof of Theorem sbcssg

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcal 3804	. . 3 ⊢ ([𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶) ↔ ∀𝑦[𝐴 / 𝑥](𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶))
2		sbcimg 3793	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶) ↔ ([𝐴 / 𝑥]𝑦 ∈ 𝐵 → [𝐴 / 𝑥]𝑦 ∈ 𝐶)))
3		sbcel2 4384	. . . . . 6 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)
4		sbcel2 4384	. . . . . 6 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)
5	3, 4	imbi12i 353	. . . . 5 ⊢ (([𝐴 / 𝑥]𝑦 ∈ 𝐵 → [𝐴 / 𝑥]𝑦 ∈ 𝐶) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))
6	2, 5	bitrdi 290	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))
7	6	albidv 1950	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑦[𝐴 / 𝑥](𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶) ↔ ∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))
8	1, 7	bitrid 286	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶) ↔ ∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))
9		df-ss 3923	. . 3 ⊢ (𝐵 ⊆ 𝐶 ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶))
10	9	sbcbii 3801	. 2 ⊢ ([𝐴 / 𝑥]𝐵 ⊆ 𝐶 ↔ [𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶))
11		df-ss 3923	. 2 ⊢ (⦋𝐴 / 𝑥⦌𝐵 ⊆ ⦋𝐴 / 𝑥⦌𝐶 ↔ ∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))
12	8, 10, 11	3bitr4g 317	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 ⊆ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ⊆ ⦋𝐴 / 𝑥⦌𝐶))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1568   ∈ wcel 2143  [wsbc 3745  ⦋csb 3854   ⊆ wss 3906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-ss 3923  df-nul 4288

This theorem is referenced by:  sbcrel  5769  sbcfg  6705  csbfrecsg  8282  iuninc  32886  minregex  44243  brtrclfv2  44436  cotrclrcl  44451  sbcheg  44488