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Theorem sbcabel 3807
 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 3459 . 2 (𝐴𝑉𝐴 ∈ V)
2 sbcex2 3781 . . . 4 ([𝐴 / 𝑥]𝑤(𝑤 = {𝑦𝜑} ∧ 𝑤𝐵) ↔ ∃𝑤[𝐴 / 𝑥](𝑤 = {𝑦𝜑} ∧ 𝑤𝐵))
3 sbcan 3768 . . . . . 6 ([𝐴 / 𝑥](𝑤 = {𝑦𝜑} ∧ 𝑤𝐵) ↔ ([𝐴 / 𝑥]𝑤 = {𝑦𝜑} ∧ [𝐴 / 𝑥]𝑤𝐵))
4 sbcal 3780 . . . . . . . . 9 ([𝐴 / 𝑥]𝑦(𝑦𝑤𝜑) ↔ ∀𝑦[𝐴 / 𝑥](𝑦𝑤𝜑))
5 sbcbig 3770 . . . . . . . . . . 11 (𝐴 ∈ V → ([𝐴 / 𝑥](𝑦𝑤𝜑) ↔ ([𝐴 / 𝑥]𝑦𝑤[𝐴 / 𝑥]𝜑)))
6 sbcg 3793 . . . . . . . . . . . 12 (𝐴 ∈ V → ([𝐴 / 𝑥]𝑦𝑤𝑦𝑤))
76bibi1d 347 . . . . . . . . . . 11 (𝐴 ∈ V → (([𝐴 / 𝑥]𝑦𝑤[𝐴 / 𝑥]𝜑) ↔ (𝑦𝑤[𝐴 / 𝑥]𝜑)))
85, 7bitrd 282 . . . . . . . . . 10 (𝐴 ∈ V → ([𝐴 / 𝑥](𝑦𝑤𝜑) ↔ (𝑦𝑤[𝐴 / 𝑥]𝜑)))
98albidv 1921 . . . . . . . . 9 (𝐴 ∈ V → (∀𝑦[𝐴 / 𝑥](𝑦𝑤𝜑) ↔ ∀𝑦(𝑦𝑤[𝐴 / 𝑥]𝜑)))
104, 9syl5bb 286 . . . . . . . 8 (𝐴 ∈ V → ([𝐴 / 𝑥]𝑦(𝑦𝑤𝜑) ↔ ∀𝑦(𝑦𝑤[𝐴 / 𝑥]𝜑)))
11 abeq2 2922 . . . . . . . . 9 (𝑤 = {𝑦𝜑} ↔ ∀𝑦(𝑦𝑤𝜑))
1211sbcbii 3776 . . . . . . . 8 ([𝐴 / 𝑥]𝑤 = {𝑦𝜑} ↔ [𝐴 / 𝑥]𝑦(𝑦𝑤𝜑))
13 abeq2 2922 . . . . . . . 8 (𝑤 = {𝑦[𝐴 / 𝑥]𝜑} ↔ ∀𝑦(𝑦𝑤[𝐴 / 𝑥]𝜑))
1410, 12, 133bitr4g 317 . . . . . . 7 (𝐴 ∈ V → ([𝐴 / 𝑥]𝑤 = {𝑦𝜑} ↔ 𝑤 = {𝑦[𝐴 / 𝑥]𝜑}))
15 sbcabel.1 . . . . . . . . 9 𝑥𝐵
1615nfcri 2943 . . . . . . . 8 𝑥 𝑤𝐵
1716sbcgf 3791 . . . . . . 7 (𝐴 ∈ V → ([𝐴 / 𝑥]𝑤𝐵𝑤𝐵))
1814, 17anbi12d 633 . . . . . 6 (𝐴 ∈ V → (([𝐴 / 𝑥]𝑤 = {𝑦𝜑} ∧ [𝐴 / 𝑥]𝑤𝐵) ↔ (𝑤 = {𝑦[𝐴 / 𝑥]𝜑} ∧ 𝑤𝐵)))
193, 18syl5bb 286 . . . . 5 (𝐴 ∈ V → ([𝐴 / 𝑥](𝑤 = {𝑦𝜑} ∧ 𝑤𝐵) ↔ (𝑤 = {𝑦[𝐴 / 𝑥]𝜑} ∧ 𝑤𝐵)))
2019exbidv 1922 . . . 4 (𝐴 ∈ V → (∃𝑤[𝐴 / 𝑥](𝑤 = {𝑦𝜑} ∧ 𝑤𝐵) ↔ ∃𝑤(𝑤 = {𝑦[𝐴 / 𝑥]𝜑} ∧ 𝑤𝐵)))
212, 20syl5bb 286 . . 3 (𝐴 ∈ V → ([𝐴 / 𝑥]𝑤(𝑤 = {𝑦𝜑} ∧ 𝑤𝐵) ↔ ∃𝑤(𝑤 = {𝑦[𝐴 / 𝑥]𝜑} ∧ 𝑤𝐵)))
22 dfclel 2871 . . . 4 ({𝑦𝜑} ∈ 𝐵 ↔ ∃𝑤(𝑤 = {𝑦𝜑} ∧ 𝑤𝐵))
2322sbcbii 3776 . . 3 ([𝐴 / 𝑥]{𝑦𝜑} ∈ 𝐵[𝐴 / 𝑥]𝑤(𝑤 = {𝑦𝜑} ∧ 𝑤𝐵))
24 dfclel 2871 . . 3 ({𝑦[𝐴 / 𝑥]𝜑} ∈ 𝐵 ↔ ∃𝑤(𝑤 = {𝑦[𝐴 / 𝑥]𝜑} ∧ 𝑤𝐵))
2521, 23, 243bitr4g 317 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]{𝑦𝜑} ∈ 𝐵 ↔ {𝑦[𝐴 / 𝑥]𝜑} ∈ 𝐵))
261, 25syl 17 1 (𝐴𝑉 → ([𝐴 / 𝑥]{𝑦𝜑} ∈ 𝐵 ↔ {𝑦[𝐴 / 𝑥]𝜑} ∈ 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2776  Ⅎwnfc 2936  Vcvv 3441  [wsbc 3720 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-v 3443  df-sbc 3721 This theorem is referenced by:  csbexg  5178
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