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Theorem sbceqg 4409
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 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))

Proof of Theorem sbceqg
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3780 . . 3 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝐵 = 𝐶[𝐴 / 𝑥]𝐵 = 𝐶))
2 dfsbcq2 3780 . . . . 5 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝑦𝐵[𝐴 / 𝑥]𝑦𝐵))
32abbidv 2800 . . . 4 (𝑧 = 𝐴 → {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵} = {𝑦[𝐴 / 𝑥]𝑦𝐵})
4 dfsbcq2 3780 . . . . 5 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐶))
54abbidv 2800 . . . 4 (𝑧 = 𝐴 → {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶} = {𝑦[𝐴 / 𝑥]𝑦𝐶})
63, 5eqeq12d 2747 . . 3 (𝑧 = 𝐴 → ({𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵} = {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶} ↔ {𝑦[𝐴 / 𝑥]𝑦𝐵} = {𝑦[𝐴 / 𝑥]𝑦𝐶}))
7 nfs1v 2152 . . . . . 6 𝑥[𝑧 / 𝑥]𝑦𝐵
87nfab 2908 . . . . 5 𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵}
9 nfs1v 2152 . . . . . 6 𝑥[𝑧 / 𝑥]𝑦𝐶
109nfab 2908 . . . . 5 𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶}
118, 10nfeq 2915 . . . 4 𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵} = {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶}
12 sbab 2881 . . . . 5 (𝑥 = 𝑧𝐵 = {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵})
13 sbab 2881 . . . . 5 (𝑥 = 𝑧𝐶 = {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶})
1412, 13eqeq12d 2747 . . . 4 (𝑥 = 𝑧 → (𝐵 = 𝐶 ↔ {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵} = {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶}))
1511, 14sbiev 2307 . . 3 ([𝑧 / 𝑥]𝐵 = 𝐶 ↔ {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵} = {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶})
161, 6, 15vtoclbg 3544 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ {𝑦[𝐴 / 𝑥]𝑦𝐵} = {𝑦[𝐴 / 𝑥]𝑦𝐶}))
17 df-csb 3894 . . 3 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
18 df-csb 3894 . . 3 𝐴 / 𝑥𝐶 = {𝑦[𝐴 / 𝑥]𝑦𝐶}
1917, 18eqeq12i 2749 . 2 (𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶 ↔ {𝑦[𝐴 / 𝑥]𝑦𝐵} = {𝑦[𝐴 / 𝑥]𝑦𝐶})
2016, 19bitr4di 289 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1540  [wsb 2066  wcel 2105  {cab 2708  [wsbc 3777  csb 3893
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-sbc 3778  df-csb 3894
This theorem is referenced by:  sbceqi  4410  sbcne12  4412  sbceq1g  4414  sbceq2g  4416  csbie2df  4440  sbcfng  6714  csbfrecsg  8275  swrdspsleq  14622  fprodmodd  15948  relowlpssretop  36709  rdgeqoa  36715  poimirlem25  36977  cdlemk42  40276  minregex  42748  onfrALTlem5  43766  onfrALTlem4  43767  csbingVD  44108  onfrALTlem5VD  44109  onfrALTlem4VD  44110  csbeq2gVD  44116  csbsngVD  44117  csbunigVD  44122  csbfv12gALTVD  44123
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