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Theorem sbabel 2359
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 2016 . . 3 ([𝑦 / 𝑥]∃𝑣(𝑣 = {𝑧𝜑} ∧ 𝑣𝐴) ↔ ∃𝑣[𝑦 / 𝑥](𝑣 = {𝑧𝜑} ∧ 𝑣𝐴))
2 sban 1967 . . . . 5 ([𝑦 / 𝑥](𝑣 = {𝑧𝜑} ∧ 𝑣𝐴) ↔ ([𝑦 / 𝑥]𝑣 = {𝑧𝜑} ∧ [𝑦 / 𝑥]𝑣𝐴))
3 nfv 1539 . . . . . . . . . 10 𝑥 𝑧𝑣
43sbf 1788 . . . . . . . . 9 ([𝑦 / 𝑥]𝑧𝑣𝑧𝑣)
54sbrbis 1973 . . . . . . . 8 ([𝑦 / 𝑥](𝑧𝑣𝜑) ↔ (𝑧𝑣 ↔ [𝑦 / 𝑥]𝜑))
65sbalv 2017 . . . . . . 7 ([𝑦 / 𝑥]∀𝑧(𝑧𝑣𝜑) ↔ ∀𝑧(𝑧𝑣 ↔ [𝑦 / 𝑥]𝜑))
7 abeq2 2298 . . . . . . . 8 (𝑣 = {𝑧𝜑} ↔ ∀𝑧(𝑧𝑣𝜑))
87sbbii 1776 . . . . . . 7 ([𝑦 / 𝑥]𝑣 = {𝑧𝜑} ↔ [𝑦 / 𝑥]∀𝑧(𝑧𝑣𝜑))
9 abeq2 2298 . . . . . . 7 (𝑣 = {𝑧 ∣ [𝑦 / 𝑥]𝜑} ↔ ∀𝑧(𝑧𝑣 ↔ [𝑦 / 𝑥]𝜑))
106, 8, 93bitr4i 212 . . . . . 6 ([𝑦 / 𝑥]𝑣 = {𝑧𝜑} ↔ 𝑣 = {𝑧 ∣ [𝑦 / 𝑥]𝜑})
11 sbabel.1 . . . . . . . 8 𝑥𝐴
1211nfcri 2326 . . . . . . 7 𝑥 𝑣𝐴
1312sbf 1788 . . . . . 6 ([𝑦 / 𝑥]𝑣𝐴𝑣𝐴)
1410, 13anbi12i 460 . . . . 5 (([𝑦 / 𝑥]𝑣 = {𝑧𝜑} ∧ [𝑦 / 𝑥]𝑣𝐴) ↔ (𝑣 = {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∧ 𝑣𝐴))
152, 14bitri 184 . . . 4 ([𝑦 / 𝑥](𝑣 = {𝑧𝜑} ∧ 𝑣𝐴) ↔ (𝑣 = {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∧ 𝑣𝐴))
1615exbii 1616 . . 3 (∃𝑣[𝑦 / 𝑥](𝑣 = {𝑧𝜑} ∧ 𝑣𝐴) ↔ ∃𝑣(𝑣 = {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∧ 𝑣𝐴))
171, 16bitri 184 . 2 ([𝑦 / 𝑥]∃𝑣(𝑣 = {𝑧𝜑} ∧ 𝑣𝐴) ↔ ∃𝑣(𝑣 = {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∧ 𝑣𝐴))
18 df-clel 2185 . . 3 ({𝑧𝜑} ∈ 𝐴 ↔ ∃𝑣(𝑣 = {𝑧𝜑} ∧ 𝑣𝐴))
1918sbbii 1776 . 2 ([𝑦 / 𝑥]{𝑧𝜑} ∈ 𝐴 ↔ [𝑦 / 𝑥]∃𝑣(𝑣 = {𝑧𝜑} ∧ 𝑣𝐴))
20 df-clel 2185 . 2 ({𝑧 ∣ [𝑦 / 𝑥]𝜑} ∈ 𝐴 ↔ ∃𝑣(𝑣 = {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∧ 𝑣𝐴))
2117, 19, 203bitr4i 212 1 ([𝑦 / 𝑥]{𝑧𝜑} ∈ 𝐴 ↔ {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∈ 𝐴)
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105  wal 1362   = wceq 1364  wex 1503  [wsb 1773  wcel 2160  {cab 2175  wnfc 2319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321
This theorem is referenced by: (None)
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