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Theorem sbcabel 2867
Description: Interchange class substitution and class abstraction. (Contributed by NM, 5-Nov-2005.)
Hypothesis
Ref Expression
sbcabel.1 𝑥𝐵
Assertion
Ref Expression
sbcabel (𝐴𝑉 → ([𝐴 / 𝑥]{𝑦𝜑} ∈ 𝐵 ↔ {𝑦[𝐴 / 𝑥]𝜑} ∈ 𝐵))
Distinct variable groups:   𝑦,𝐴   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem sbcabel
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 elex 2583 . 2 (𝐴𝑉𝐴 ∈ V)
2 sbcexg 2840 . . . 4 (𝐴 ∈ V → ([𝐴 / 𝑥]𝑤(𝑤 = {𝑦𝜑} ∧ 𝑤𝐵) ↔ ∃𝑤[𝐴 / 𝑥](𝑤 = {𝑦𝜑} ∧ 𝑤𝐵)))
3 sbcang 2829 . . . . . 6 (𝐴 ∈ V → ([𝐴 / 𝑥](𝑤 = {𝑦𝜑} ∧ 𝑤𝐵) ↔ ([𝐴 / 𝑥]𝑤 = {𝑦𝜑} ∧ [𝐴 / 𝑥]𝑤𝐵)))
4 sbcalg 2838 . . . . . . . . 9 (𝐴 ∈ V → ([𝐴 / 𝑥]𝑦(𝑦𝑤𝜑) ↔ ∀𝑦[𝐴 / 𝑥](𝑦𝑤𝜑)))
5 sbcbig 2832 . . . . . . . . . . 11 (𝐴 ∈ V → ([𝐴 / 𝑥](𝑦𝑤𝜑) ↔ ([𝐴 / 𝑥]𝑦𝑤[𝐴 / 𝑥]𝜑)))
6 sbcg 2855 . . . . . . . . . . . 12 (𝐴 ∈ V → ([𝐴 / 𝑥]𝑦𝑤𝑦𝑤))
76bibi1d 226 . . . . . . . . . . 11 (𝐴 ∈ V → (([𝐴 / 𝑥]𝑦𝑤[𝐴 / 𝑥]𝜑) ↔ (𝑦𝑤[𝐴 / 𝑥]𝜑)))
85, 7bitrd 181 . . . . . . . . . 10 (𝐴 ∈ V → ([𝐴 / 𝑥](𝑦𝑤𝜑) ↔ (𝑦𝑤[𝐴 / 𝑥]𝜑)))
98albidv 1721 . . . . . . . . 9 (𝐴 ∈ V → (∀𝑦[𝐴 / 𝑥](𝑦𝑤𝜑) ↔ ∀𝑦(𝑦𝑤[𝐴 / 𝑥]𝜑)))
104, 9bitrd 181 . . . . . . . 8 (𝐴 ∈ V → ([𝐴 / 𝑥]𝑦(𝑦𝑤𝜑) ↔ ∀𝑦(𝑦𝑤[𝐴 / 𝑥]𝜑)))
11 abeq2 2162 . . . . . . . . 9 (𝑤 = {𝑦𝜑} ↔ ∀𝑦(𝑦𝑤𝜑))
1211sbcbii 2845 . . . . . . . 8 ([𝐴 / 𝑥]𝑤 = {𝑦𝜑} ↔ [𝐴 / 𝑥]𝑦(𝑦𝑤𝜑))
13 abeq2 2162 . . . . . . . 8 (𝑤 = {𝑦[𝐴 / 𝑥]𝜑} ↔ ∀𝑦(𝑦𝑤[𝐴 / 𝑥]𝜑))
1410, 12, 133bitr4g 216 . . . . . . 7 (𝐴 ∈ V → ([𝐴 / 𝑥]𝑤 = {𝑦𝜑} ↔ 𝑤 = {𝑦[𝐴 / 𝑥]𝜑}))
15 sbcabel.1 . . . . . . . . 9 𝑥𝐵
1615nfcri 2188 . . . . . . . 8 𝑥 𝑤𝐵
1716sbcgf 2853 . . . . . . 7 (𝐴 ∈ V → ([𝐴 / 𝑥]𝑤𝐵𝑤𝐵))
1814, 17anbi12d 450 . . . . . 6 (𝐴 ∈ V → (([𝐴 / 𝑥]𝑤 = {𝑦𝜑} ∧ [𝐴 / 𝑥]𝑤𝐵) ↔ (𝑤 = {𝑦[𝐴 / 𝑥]𝜑} ∧ 𝑤𝐵)))
193, 18bitrd 181 . . . . 5 (𝐴 ∈ V → ([𝐴 / 𝑥](𝑤 = {𝑦𝜑} ∧ 𝑤𝐵) ↔ (𝑤 = {𝑦[𝐴 / 𝑥]𝜑} ∧ 𝑤𝐵)))
2019exbidv 1722 . . . 4 (𝐴 ∈ V → (∃𝑤[𝐴 / 𝑥](𝑤 = {𝑦𝜑} ∧ 𝑤𝐵) ↔ ∃𝑤(𝑤 = {𝑦[𝐴 / 𝑥]𝜑} ∧ 𝑤𝐵)))
212, 20bitrd 181 . . 3 (𝐴 ∈ V → ([𝐴 / 𝑥]𝑤(𝑤 = {𝑦𝜑} ∧ 𝑤𝐵) ↔ ∃𝑤(𝑤 = {𝑦[𝐴 / 𝑥]𝜑} ∧ 𝑤𝐵)))
22 df-clel 2052 . . . 4 ({𝑦𝜑} ∈ 𝐵 ↔ ∃𝑤(𝑤 = {𝑦𝜑} ∧ 𝑤𝐵))
2322sbcbii 2845 . . 3 ([𝐴 / 𝑥]{𝑦𝜑} ∈ 𝐵[𝐴 / 𝑥]𝑤(𝑤 = {𝑦𝜑} ∧ 𝑤𝐵))
24 df-clel 2052 . . 3 ({𝑦[𝐴 / 𝑥]𝜑} ∈ 𝐵 ↔ ∃𝑤(𝑤 = {𝑦[𝐴 / 𝑥]𝜑} ∧ 𝑤𝐵))
2521, 23, 243bitr4g 216 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]{𝑦𝜑} ∈ 𝐵 ↔ {𝑦[𝐴 / 𝑥]𝜑} ∈ 𝐵))
261, 25syl 14 1 (𝐴𝑉 → ([𝐴 / 𝑥]{𝑦𝜑} ∈ 𝐵 ↔ {𝑦[𝐴 / 𝑥]𝜑} ∈ 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1257   = wceq 1259  wex 1397  wcel 1409  {cab 2042  wnfc 2181  Vcvv 2574  [wsbc 2787
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-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-sbc 2788
This theorem is referenced by:  csbexga  3913
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