Theorem sbceqg 4378

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 3759	. . 3 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝐵 = 𝐶 ↔ [𝐴 / 𝑥]𝐵 = 𝐶))
2		dfsbcq2 3759	. . . . 5 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝑦 ∈ 𝐵 ↔ [𝐴 / 𝑥]𝑦 ∈ 𝐵))
3	2	abbidv 2796	. . . 4 ⊢ (𝑧 = 𝐴 → {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵})
4		dfsbcq2 3759	. . . . 5 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝑦 ∈ 𝐶 ↔ [𝐴 / 𝑥]𝑦 ∈ 𝐶))
5	4	abbidv 2796	. . . 4 ⊢ (𝑧 = 𝐴 → {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶})
6	3, 5	eqeq12d 2746	. . 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 2157	. . . . . 6 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝑦 ∈ 𝐵
8	7	nfab 2898	. . . . 5 ⊢ Ⅎ𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵}
9		nfs1v 2157	. . . . . 6 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝑦 ∈ 𝐶
10	9	nfab 2898	. . . . 5 ⊢ Ⅎ𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶}
11	8, 10	nfeq 2906	. . . 4 ⊢ Ⅎ𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶}
12		sbab 2876	. . . . 5 ⊢ (𝑥 = 𝑧 → 𝐵 = {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵})
13		sbab 2876	. . . . 5 ⊢ (𝑥 = 𝑧 → 𝐶 = {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶})
14	12, 13	eqeq12d 2746	. . . 4 ⊢ (𝑥 = 𝑧 → (𝐵 = 𝐶 ↔ {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶}))
15	11, 14	sbiev 2313	. . 3 ⊢ ([𝑧 / 𝑥]𝐵 = 𝐶 ↔ {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶})
16	1, 6, 15	vtoclbg 3526	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶}))
17		df-csb 3866	. . 3 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
18		df-csb 3866	. . 3 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶}
19	17, 18	eqeq12i 2748	. 2 ⊢ (⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶 ↔ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶})
20	16, 19	bitr4di 289	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540  [wsb 2065   ∈ wcel 2109  {cab 2708  [wsbc 3756  ⦋csb 3865

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-sbc 3757  df-csb 3866

This theorem is referenced by:  sbceqi  4379  sbcne12  4381  sbceq1g  4383  sbceq2g  4385  csbie2df  4409  sbcfng  6688  csbfrecsg  8266  swrdspsleq  14637  fprodmodd  15970  relowlpssretop  37359  rdgeqoa  37365  poimirlem25  37646  cdlemk42  40942  minregex  43530  onfrALTlem5  44539  onfrALTlem4  44540  csbingVD  44880  onfrALTlem5VD  44881  onfrALTlem4VD  44882  csbeq2gVD  44888  csbsngVD  44889  csbunigVD  44894  csbfv12gALTVD  44895