Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  sbceqg GIF version

Theorem sbceqg 2866
 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 2767 . . 3 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝐵 = 𝐶[𝐴 / 𝑥]𝐵 = 𝐶))
2 dfsbcq2 2767 . . . . 5 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝑦𝐵[𝐴 / 𝑥]𝑦𝐵))
32abbidv 2155 . . . 4 (𝑧 = 𝐴 → {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵} = {𝑦[𝐴 / 𝑥]𝑦𝐵})
4 dfsbcq2 2767 . . . . 5 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐶))
54abbidv 2155 . . . 4 (𝑧 = 𝐴 → {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶} = {𝑦[𝐴 / 𝑥]𝑦𝐶})
63, 5eqeq12d 2054 . . 3 (𝑧 = 𝐴 → ({𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵} = {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶} ↔ {𝑦[𝐴 / 𝑥]𝑦𝐵} = {𝑦[𝐴 / 𝑥]𝑦𝐶}))
7 nfs1v 1815 . . . . . 6 𝑥[𝑧 / 𝑥]𝑦𝐵
87nfab 2182 . . . . 5 𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵}
9 nfs1v 1815 . . . . . 6 𝑥[𝑧 / 𝑥]𝑦𝐶
109nfab 2182 . . . . 5 𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶}
118, 10nfeq 2185 . . . 4 𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵} = {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶}
12 sbab 2164 . . . . 5 (𝑥 = 𝑧𝐵 = {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵})
13 sbab 2164 . . . . 5 (𝑥 = 𝑧𝐶 = {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶})
1412, 13eqeq12d 2054 . . . 4 (𝑥 = 𝑧 → (𝐵 = 𝐶 ↔ {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵} = {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶}))
1511, 14sbie 1674 . . 3 ([𝑧 / 𝑥]𝐵 = 𝐶 ↔ {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵} = {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶})
161, 6, 15vtoclbg 2614 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ {𝑦[𝐴 / 𝑥]𝑦𝐵} = {𝑦[𝐴 / 𝑥]𝑦𝐶}))
17 df-csb 2853 . . 3 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
18 df-csb 2853 . . 3 𝐴 / 𝑥𝐶 = {𝑦[𝐴 / 𝑥]𝑦𝐶}
1917, 18eqeq12i 2053 . 2 (𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶 ↔ {𝑦[𝐴 / 𝑥]𝑦𝐵} = {𝑦[𝐴 / 𝑥]𝑦𝐶})
2016, 19syl6bbr 187 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1243   ∈ wcel 1393  [wsb 1645  {cab 2026  [wsbc 2764  ⦋csb 2852 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765  df-csb 2853 This theorem is referenced by:  sbcne12g  2868  sbceq1g  2870  sbceq2g  2872  sbcfng  5044
 Copyright terms: Public domain W3C validator