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Theorem sbabel 2307
 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 1979 . . 3 ([𝑦 / 𝑥]∃𝑣(𝑣 = {𝑧𝜑} ∧ 𝑣𝐴) ↔ ∃𝑣[𝑦 / 𝑥](𝑣 = {𝑧𝜑} ∧ 𝑣𝐴))
2 sban 1928 . . . . 5 ([𝑦 / 𝑥](𝑣 = {𝑧𝜑} ∧ 𝑣𝐴) ↔ ([𝑦 / 𝑥]𝑣 = {𝑧𝜑} ∧ [𝑦 / 𝑥]𝑣𝐴))
3 nfv 1508 . . . . . . . . . 10 𝑥 𝑧𝑣
43sbf 1750 . . . . . . . . 9 ([𝑦 / 𝑥]𝑧𝑣𝑧𝑣)
54sbrbis 1934 . . . . . . . 8 ([𝑦 / 𝑥](𝑧𝑣𝜑) ↔ (𝑧𝑣 ↔ [𝑦 / 𝑥]𝜑))
65sbalv 1980 . . . . . . 7 ([𝑦 / 𝑥]∀𝑧(𝑧𝑣𝜑) ↔ ∀𝑧(𝑧𝑣 ↔ [𝑦 / 𝑥]𝜑))
7 abeq2 2248 . . . . . . . 8 (𝑣 = {𝑧𝜑} ↔ ∀𝑧(𝑧𝑣𝜑))
87sbbii 1738 . . . . . . 7 ([𝑦 / 𝑥]𝑣 = {𝑧𝜑} ↔ [𝑦 / 𝑥]∀𝑧(𝑧𝑣𝜑))
9 abeq2 2248 . . . . . . 7 (𝑣 = {𝑧 ∣ [𝑦 / 𝑥]𝜑} ↔ ∀𝑧(𝑧𝑣 ↔ [𝑦 / 𝑥]𝜑))
106, 8, 93bitr4i 211 . . . . . 6 ([𝑦 / 𝑥]𝑣 = {𝑧𝜑} ↔ 𝑣 = {𝑧 ∣ [𝑦 / 𝑥]𝜑})
11 sbabel.1 . . . . . . . 8 𝑥𝐴
1211nfcri 2275 . . . . . . 7 𝑥 𝑣𝐴
1312sbf 1750 . . . . . 6 ([𝑦 / 𝑥]𝑣𝐴𝑣𝐴)
1410, 13anbi12i 455 . . . . 5 (([𝑦 / 𝑥]𝑣 = {𝑧𝜑} ∧ [𝑦 / 𝑥]𝑣𝐴) ↔ (𝑣 = {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∧ 𝑣𝐴))
152, 14bitri 183 . . . 4 ([𝑦 / 𝑥](𝑣 = {𝑧𝜑} ∧ 𝑣𝐴) ↔ (𝑣 = {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∧ 𝑣𝐴))
1615exbii 1584 . . 3 (∃𝑣[𝑦 / 𝑥](𝑣 = {𝑧𝜑} ∧ 𝑣𝐴) ↔ ∃𝑣(𝑣 = {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∧ 𝑣𝐴))
171, 16bitri 183 . 2 ([𝑦 / 𝑥]∃𝑣(𝑣 = {𝑧𝜑} ∧ 𝑣𝐴) ↔ ∃𝑣(𝑣 = {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∧ 𝑣𝐴))
18 df-clel 2135 . . 3 ({𝑧𝜑} ∈ 𝐴 ↔ ∃𝑣(𝑣 = {𝑧𝜑} ∧ 𝑣𝐴))
1918sbbii 1738 . 2 ([𝑦 / 𝑥]{𝑧𝜑} ∈ 𝐴 ↔ [𝑦 / 𝑥]∃𝑣(𝑣 = {𝑧𝜑} ∧ 𝑣𝐴))
20 df-clel 2135 . 2 ({𝑧 ∣ [𝑦 / 𝑥]𝜑} ∈ 𝐴 ↔ ∃𝑣(𝑣 = {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∧ 𝑣𝐴))
2117, 19, 203bitr4i 211 1 ([𝑦 / 𝑥]{𝑧𝜑} ∈ 𝐴 ↔ {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∈ 𝐴)
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   ↔ wb 104  ∀wal 1329   = wceq 1331  ∃wex 1468   ∈ wcel 1480  [wsb 1735  {cab 2125  Ⅎwnfc 2268 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270 This theorem is referenced by: (None)
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