Theorem sbcel12gOLD 38754

Description: Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) Obsolete as of 18-Aug-2018. Use sbcel12 3983 instead. (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
sbcel12gOLD	⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶))

Proof of Theorem sbcel12gOLD

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 3438	. . 3 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝐵 ∈ 𝐶 ↔ [𝐴 / 𝑥]𝐵 ∈ 𝐶))
2		dfsbcq2 3438	. . . . 5 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝑦 ∈ 𝐵 ↔ [𝐴 / 𝑥]𝑦 ∈ 𝐵))
3	2	abbidv 2741	. . . 4 ⊢ (𝑧 = 𝐴 → {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵})
4		dfsbcq2 3438	. . . . 5 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝑦 ∈ 𝐶 ↔ [𝐴 / 𝑥]𝑦 ∈ 𝐶))
5	4	abbidv 2741	. . . 4 ⊢ (𝑧 = 𝐴 → {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶})
6	3, 5	eleq12d 2695	. . 3 ⊢ (𝑧 = 𝐴 → ({𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵} ∈ {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶} ↔ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} ∈ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶}))
7		nfs1v 2437	. . . . . 6 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝑦 ∈ 𝐵
8	7	nfab 2769	. . . . 5 ⊢ Ⅎ𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵}
9		nfs1v 2437	. . . . . 6 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝑦 ∈ 𝐶
10	9	nfab 2769	. . . . 5 ⊢ Ⅎ𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶}
11	8, 10	nfel 2777	. . . 4 ⊢ Ⅎ𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵} ∈ {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶}
12		sbab 2750	. . . . 5 ⊢ (𝑥 = 𝑧 → 𝐵 = {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵})
13		sbab 2750	. . . . 5 ⊢ (𝑥 = 𝑧 → 𝐶 = {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶})
14	12, 13	eleq12d 2695	. . . 4 ⊢ (𝑥 = 𝑧 → (𝐵 ∈ 𝐶 ↔ {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵} ∈ {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶}))
15	11, 14	sbie 2408	. . 3 ⊢ ([𝑧 / 𝑥]𝐵 ∈ 𝐶 ↔ {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵} ∈ {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶})
16	1, 6, 15	vtoclbg 3267	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} ∈ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶}))
17		df-csb 3534	. . 3 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
18		df-csb 3534	. . 3 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶}
19	17, 18	eleq12i 2694	. 2 ⊢ (⦋𝐴 / 𝑥⦌𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶 ↔ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} ∈ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶})
20	16, 19	syl6bbr 278	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶))

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1483  [wsb 1880   ∈ wcel 1990  {cab 2608  [wsbc 3435  ⦋csb 3533

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  df-csb 3534

This theorem is referenced by:  sbcel2gOLD  38755  csbxpgVD  39130  csbrngVD  39132