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Theorem sbcel12g 2986
 Description: Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
sbcel12g (𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))

Proof of Theorem sbcel12g
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 2883 . . 3 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝐵𝐶[𝐴 / 𝑥]𝐵𝐶))
2 dfsbcq2 2883 . . . . 5 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝑦𝐵[𝐴 / 𝑥]𝑦𝐵))
32abbidv 2233 . . . 4 (𝑧 = 𝐴 → {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵} = {𝑦[𝐴 / 𝑥]𝑦𝐵})
4 dfsbcq2 2883 . . . . 5 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐶))
54abbidv 2233 . . . 4 (𝑧 = 𝐴 → {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶} = {𝑦[𝐴 / 𝑥]𝑦𝐶})
63, 5eleq12d 2186 . . 3 (𝑧 = 𝐴 → ({𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵} ∈ {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶} ↔ {𝑦[𝐴 / 𝑥]𝑦𝐵} ∈ {𝑦[𝐴 / 𝑥]𝑦𝐶}))
7 nfs1v 1890 . . . . . 6 𝑥[𝑧 / 𝑥]𝑦𝐵
87nfab 2261 . . . . 5 𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵}
9 nfs1v 1890 . . . . . 6 𝑥[𝑧 / 𝑥]𝑦𝐶
109nfab 2261 . . . . 5 𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶}
118, 10nfel 2265 . . . 4 𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵} ∈ {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶}
12 sbab 2242 . . . . 5 (𝑥 = 𝑧𝐵 = {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵})
13 sbab 2242 . . . . 5 (𝑥 = 𝑧𝐶 = {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶})
1412, 13eleq12d 2186 . . . 4 (𝑥 = 𝑧 → (𝐵𝐶 ↔ {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵} ∈ {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶}))
1511, 14sbie 1747 . . 3 ([𝑧 / 𝑥]𝐵𝐶 ↔ {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵} ∈ {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶})
161, 6, 15vtoclbg 2719 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶 ↔ {𝑦[𝐴 / 𝑥]𝑦𝐵} ∈ {𝑦[𝐴 / 𝑥]𝑦𝐶}))
17 df-csb 2974 . . 3 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
18 df-csb 2974 . . 3 𝐴 / 𝑥𝐶 = {𝑦[𝐴 / 𝑥]𝑦𝐶}
1917, 18eleq12i 2183 . 2 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶 ↔ {𝑦[𝐴 / 𝑥]𝑦𝐵} ∈ {𝑦[𝐴 / 𝑥]𝑦𝐶})
2016, 19syl6bbr 197 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   = wceq 1314   ∈ wcel 1463  [wsb 1718  {cab 2101  [wsbc 2880  ⦋csb 2973 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097 This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-sbc 2881  df-csb 2974 This theorem is referenced by:  sbcnel12g  2988  sbcel1g  2990  sbcel2g  2992  sbccsb2g  3000  ixpsnval  6561
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