Theorem sbceqg 3936

Description: Distribute proper substitution through an equality relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Assertion

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

Proof of Theorem sbceqg

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

Step	Hyp	Ref	Expression
1		dfsbcq2 3405	. . 3 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝐵 = 𝐶 ↔ [𝐴 / 𝑥]𝐵 = 𝐶))
2		dfsbcq2 3405	. . . . 5 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝑦 ∈ 𝐵 ↔ [𝐴 / 𝑥]𝑦 ∈ 𝐵))
3	2	abbidv 2728	. . . 4 ⊢ (𝑧 = 𝐴 → {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵})
4		dfsbcq2 3405	. . . . 5 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝑦 ∈ 𝐶 ↔ [𝐴 / 𝑥]𝑦 ∈ 𝐶))
5	4	abbidv 2728	. . . 4 ⊢ (𝑧 = 𝐴 → {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶})
6	3, 5	eqeq12d 2625	. . 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 2425	. . . . . 6 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝑦 ∈ 𝐵
8	7	nfab 2755	. . . . 5 ⊢ Ⅎ𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵}
9		nfs1v 2425	. . . . . 6 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝑦 ∈ 𝐶
10	9	nfab 2755	. . . . 5 ⊢ Ⅎ𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶}
11	8, 10	nfeq 2762	. . . 4 ⊢ Ⅎ𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶}
12		sbab 2737	. . . . 5 ⊢ (𝑥 = 𝑧 → 𝐵 = {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵})
13		sbab 2737	. . . . 5 ⊢ (𝑥 = 𝑧 → 𝐶 = {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶})
14	12, 13	eqeq12d 2625	. . . 4 ⊢ (𝑥 = 𝑧 → (𝐵 = 𝐶 ↔ {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶}))
15	11, 14	sbie 2396	. . 3 ⊢ ([𝑧 / 𝑥]𝐵 = 𝐶 ↔ {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶})
16	1, 6, 15	vtoclbg 3240	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶}))
17		df-csb 3500	. . 3 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
18		df-csb 3500	. . 3 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶}
19	17, 18	eqeq12i 2624	. 2 ⊢ (⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶 ↔ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶})
20	16, 19	syl6bbr 277	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶))

Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475  [wsb 1867   ∈ wcel 1977  {cab 2596  [wsbc 3402  ⦋csb 3499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-sbc 3403  df-csb 3500

This theorem is referenced by:  sbcne12  3938  sbceq1g  3940  sbceq2g  3942  sbcfng  5955  swrdspsleq  13301  fprodmodd  14567  csbwrecsg  32349  relowlpssretop  32388  rdgeqoa  32394  poimirlem25  32604  sbceqi  33083  cdlemk42  35247  onfrALTlem5  37778  onfrALTlem4  37779  csbeq2gOLD  37786  csbfv12gALTOLD  38074  csbingVD  38142  onfrALTlem5VD  38143  onfrALTlem4VD  38144  csbeq2gVD  38150  csbsngVD  38151  csbunigVD  38156  csbfv12gALTVD  38157