Theorem sbceqg 4373

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 3745	. . 3 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝐵 = 𝐶 ↔ [𝐴 / 𝑥]𝐵 = 𝐶))
2		dfsbcq2 3745	. . . . 5 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝑦 ∈ 𝐵 ↔ [𝐴 / 𝑥]𝑦 ∈ 𝐵))
3	2	abbidv 2828	. . . 4 ⊢ (𝑧 = 𝐴 → {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵})
4		dfsbcq2 3745	. . . . 5 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝑦 ∈ 𝐶 ↔ [𝐴 / 𝑥]𝑦 ∈ 𝐶))
5	4	abbidv 2828	. . . 4 ⊢ (𝑧 = 𝐴 → {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶})
6	3, 5	eqeq12d 2778	. . 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 2193	. . . . . 6 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝑦 ∈ 𝐵
8	7	nfab 2930	. . . . 5 ⊢ Ⅎ𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵}
9		nfs1v 2193	. . . . . 6 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝑦 ∈ 𝐶
10	9	nfab 2930	. . . . 5 ⊢ Ⅎ𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶}
11	8, 10	nfeq 2937	. . . 4 ⊢ Ⅎ𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶}
12		sbab 2908	. . . . 5 ⊢ (𝑥 = 𝑧 → 𝐵 = {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵})
13		sbab 2908	. . . . 5 ⊢ (𝑥 = 𝑧 → 𝐶 = {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶})
14	12, 13	eqeq12d 2778	. . . 4 ⊢ (𝑥 = 𝑧 → (𝐵 = 𝐶 ↔ {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶}))
15	11, 14	sbiev 2347	. . 3 ⊢ ([𝑧 / 𝑥]𝐵 = 𝐶 ↔ {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶})
16	1, 6, 15	vtoclbg 3522	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶}))
17		df-csb 3851	. . 3 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
18		df-csb 3851	. . 3 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶}
19	17, 18	eqeq12i 2780	. 2 ⊢ (⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶 ↔ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶})
20	16, 19	bitr4di 292	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  [wsb 2099   ∈ wcel 2145  {cab 2740  [wsbc 3742  ⦋csb 3850

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-sbc 3743  df-csb 3851

This theorem is used by:  sbceqi  4374  sbcne12  4376  sbceq1g  4378  sbceq2g  4380  csbie2df  4404  sbcfng  6703  csbfrecsg  8286  swrdspsleq  14737  fprodmodd  16088  relowlpssretop  38105  rdgeqoa  38111  poimirlem25  38381  cdlemk42  41801  minregex  44361  onfrALTlem5  45352  onfrALTlem4  45353  csbingVD  45693  onfrALTlem5VD  45694  onfrALTlem4VD  45695  csbeq2gVD  45701  csbsngVD  45702  csbunigVD  45707  csbfv12gALTVD  45708