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

Theorem sbabel 2219
Description: Theorem to move a substitution in and out of a class abstraction. (Contributed by NM, 27-Sep-2003.) (Revised by Mario Carneiro, 7-Oct-2016.)
Hypothesis
Ref Expression
sbabel.1 𝑥𝐴
Assertion
Ref Expression
sbabel ([𝑦 / 𝑥]{𝑧𝜑} ∈ 𝐴 ↔ {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∈ 𝐴)
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐴(𝑥,𝑦,𝑧)

Proof of Theorem sbabel
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 sbex 1896 . . 3 ([𝑦 / 𝑥]∃𝑣(𝑣 = {𝑧𝜑} ∧ 𝑣𝐴) ↔ ∃𝑣[𝑦 / 𝑥](𝑣 = {𝑧𝜑} ∧ 𝑣𝐴))
2 sban 1845 . . . . 5 ([𝑦 / 𝑥](𝑣 = {𝑧𝜑} ∧ 𝑣𝐴) ↔ ([𝑦 / 𝑥]𝑣 = {𝑧𝜑} ∧ [𝑦 / 𝑥]𝑣𝐴))
3 nfv 1437 . . . . . . . . . 10 𝑥 𝑧𝑣
43sbf 1676 . . . . . . . . 9 ([𝑦 / 𝑥]𝑧𝑣𝑧𝑣)
54sbrbis 1851 . . . . . . . 8 ([𝑦 / 𝑥](𝑧𝑣𝜑) ↔ (𝑧𝑣 ↔ [𝑦 / 𝑥]𝜑))
65sbalv 1897 . . . . . . 7 ([𝑦 / 𝑥]∀𝑧(𝑧𝑣𝜑) ↔ ∀𝑧(𝑧𝑣 ↔ [𝑦 / 𝑥]𝜑))
7 abeq2 2162 . . . . . . . 8 (𝑣 = {𝑧𝜑} ↔ ∀𝑧(𝑧𝑣𝜑))
87sbbii 1664 . . . . . . 7 ([𝑦 / 𝑥]𝑣 = {𝑧𝜑} ↔ [𝑦 / 𝑥]∀𝑧(𝑧𝑣𝜑))
9 abeq2 2162 . . . . . . 7 (𝑣 = {𝑧 ∣ [𝑦 / 𝑥]𝜑} ↔ ∀𝑧(𝑧𝑣 ↔ [𝑦 / 𝑥]𝜑))
106, 8, 93bitr4i 205 . . . . . 6 ([𝑦 / 𝑥]𝑣 = {𝑧𝜑} ↔ 𝑣 = {𝑧 ∣ [𝑦 / 𝑥]𝜑})
11 sbabel.1 . . . . . . . 8 𝑥𝐴
1211nfcri 2188 . . . . . . 7 𝑥 𝑣𝐴
1312sbf 1676 . . . . . 6 ([𝑦 / 𝑥]𝑣𝐴𝑣𝐴)
1410, 13anbi12i 441 . . . . 5 (([𝑦 / 𝑥]𝑣 = {𝑧𝜑} ∧ [𝑦 / 𝑥]𝑣𝐴) ↔ (𝑣 = {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∧ 𝑣𝐴))
152, 14bitri 177 . . . 4 ([𝑦 / 𝑥](𝑣 = {𝑧𝜑} ∧ 𝑣𝐴) ↔ (𝑣 = {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∧ 𝑣𝐴))
1615exbii 1512 . . 3 (∃𝑣[𝑦 / 𝑥](𝑣 = {𝑧𝜑} ∧ 𝑣𝐴) ↔ ∃𝑣(𝑣 = {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∧ 𝑣𝐴))
171, 16bitri 177 . 2 ([𝑦 / 𝑥]∃𝑣(𝑣 = {𝑧𝜑} ∧ 𝑣𝐴) ↔ ∃𝑣(𝑣 = {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∧ 𝑣𝐴))
18 df-clel 2052 . . 3 ({𝑧𝜑} ∈ 𝐴 ↔ ∃𝑣(𝑣 = {𝑧𝜑} ∧ 𝑣𝐴))
1918sbbii 1664 . 2 ([𝑦 / 𝑥]{𝑧𝜑} ∈ 𝐴 ↔ [𝑦 / 𝑥]∃𝑣(𝑣 = {𝑧𝜑} ∧ 𝑣𝐴))
20 df-clel 2052 . 2 ({𝑧 ∣ [𝑦 / 𝑥]𝜑} ∈ 𝐴 ↔ ∃𝑣(𝑣 = {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∧ 𝑣𝐴))
2117, 19, 203bitr4i 205 1 ([𝑦 / 𝑥]{𝑧𝜑} ∈ 𝐴 ↔ {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∈ 𝐴)
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102  wal 1257   = wceq 1259  wex 1397  wcel 1409  [wsb 1661  {cab 2042  wnfc 2181
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator