Theorem sbcssg 4085

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 3485	. . 3 ⊢ ([𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶) ↔ ∀𝑦[𝐴 / 𝑥](𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶))
2		sbcimg 3477	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶) ↔ ([𝐴 / 𝑥]𝑦 ∈ 𝐵 → [𝐴 / 𝑥]𝑦 ∈ 𝐶)))
3		sbcel2 3989	. . . . . 6 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵)
4		sbcel2 3989	. . . . . 6 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)
5	3, 4	imbi12i 340	. . . . 5 ⊢ (([𝐴 / 𝑥]𝑦 ∈ 𝐵 → [𝐴 / 𝑥]𝑦 ∈ 𝐶) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))
6	2, 5	syl6bb 276	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))
7	6	albidv 1849	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑦[𝐴 / 𝑥](𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶) ↔ ∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))
8	1, 7	syl5bb 272	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶) ↔ ∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))
9		dfss2 3591	. . 3 ⊢ (𝐵 ⊆ 𝐶 ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶))
10	9	sbcbii 3491	. 2 ⊢ ([𝐴 / 𝑥]𝐵 ⊆ 𝐶 ↔ [𝐴 / 𝑥]∀𝑦(𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶))
11		dfss2 3591	. 2 ⊢ (⦋𝐴 / 𝑥⦌𝐵 ⊆ ⦋𝐴 / 𝑥⦌𝐶 ↔ ∀𝑦(𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵 → 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))
12	8, 10, 11	3bitr4g 303	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 ⊆ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ⊆ ⦋𝐴 / 𝑥⦌𝐶))

Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1481   ∈ wcel 1990  [wsbc 3435  ⦋csb 3533   ⊆ wss 3574

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-fal 1489  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  df-csb 3534  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916

This theorem is referenced by:  sbcrel  5205  sbcfg  6043  iuninc  29379  csbwrecsg  33173  brtrclfv2  38019  cotrclrcl  38034  sbcheg  38073