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Theorem sbabel 2266
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 1940 . . 3 ([𝑦 / 𝑥]∃𝑣(𝑣 = {𝑧𝜑} ∧ 𝑣𝐴) ↔ ∃𝑣[𝑦 / 𝑥](𝑣 = {𝑧𝜑} ∧ 𝑣𝐴))
2 sban 1889 . . . . 5 ([𝑦 / 𝑥](𝑣 = {𝑧𝜑} ∧ 𝑣𝐴) ↔ ([𝑦 / 𝑥]𝑣 = {𝑧𝜑} ∧ [𝑦 / 𝑥]𝑣𝐴))
3 nfv 1476 . . . . . . . . . 10 𝑥 𝑧𝑣
43sbf 1718 . . . . . . . . 9 ([𝑦 / 𝑥]𝑧𝑣𝑧𝑣)
54sbrbis 1895 . . . . . . . 8 ([𝑦 / 𝑥](𝑧𝑣𝜑) ↔ (𝑧𝑣 ↔ [𝑦 / 𝑥]𝜑))
65sbalv 1941 . . . . . . 7 ([𝑦 / 𝑥]∀𝑧(𝑧𝑣𝜑) ↔ ∀𝑧(𝑧𝑣 ↔ [𝑦 / 𝑥]𝜑))
7 abeq2 2208 . . . . . . . 8 (𝑣 = {𝑧𝜑} ↔ ∀𝑧(𝑧𝑣𝜑))
87sbbii 1706 . . . . . . 7 ([𝑦 / 𝑥]𝑣 = {𝑧𝜑} ↔ [𝑦 / 𝑥]∀𝑧(𝑧𝑣𝜑))
9 abeq2 2208 . . . . . . 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 2234 . . . . . . 7 𝑥 𝑣𝐴
1312sbf 1718 . . . . . 6 ([𝑦 / 𝑥]𝑣𝐴𝑣𝐴)
1410, 13anbi12i 451 . . . . 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 1552 . . 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 2096 . . 3 ({𝑧𝜑} ∈ 𝐴 ↔ ∃𝑣(𝑣 = {𝑧𝜑} ∧ 𝑣𝐴))
1918sbbii 1706 . 2 ([𝑦 / 𝑥]{𝑧𝜑} ∈ 𝐴 ↔ [𝑦 / 𝑥]∃𝑣(𝑣 = {𝑧𝜑} ∧ 𝑣𝐴))
20 df-clel 2096 . 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 1297   = wceq 1299  wex 1436  wcel 1448  [wsb 1703  {cab 2086  wnfc 2227
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229
This theorem is referenced by: (None)
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