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Theorem sbabel 2977
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.) (Proof shortened by Wolf Lammen, 26-Dec-2019.)
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 2623 . . 3 ([𝑦 / 𝑥]∃𝑣(𝑣𝐴 ∧ ∀𝑧(𝑧𝑣𝜑)) ↔ ∃𝑣[𝑦 / 𝑥](𝑣𝐴 ∧ ∀𝑧(𝑧𝑣𝜑)))
2 sban 2558 . . . . 5 ([𝑦 / 𝑥](𝑣𝐴 ∧ ∀𝑧(𝑧𝑣𝜑)) ↔ ([𝑦 / 𝑥]𝑣𝐴 ∧ [𝑦 / 𝑥]∀𝑧(𝑧𝑣𝜑)))
3 sbabel.1 . . . . . . . 8 𝑥𝐴
43nfcri 2942 . . . . . . 7 𝑥 𝑣𝐴
54sbf 2539 . . . . . 6 ([𝑦 / 𝑥]𝑣𝐴𝑣𝐴)
6 nfv 2005 . . . . . . . . 9 𝑥 𝑧𝑣
76sbf 2539 . . . . . . . 8 ([𝑦 / 𝑥]𝑧𝑣𝑧𝑣)
87sbrbis 2564 . . . . . . 7 ([𝑦 / 𝑥](𝑧𝑣𝜑) ↔ (𝑧𝑣 ↔ [𝑦 / 𝑥]𝜑))
98sbalv 2624 . . . . . 6 ([𝑦 / 𝑥]∀𝑧(𝑧𝑣𝜑) ↔ ∀𝑧(𝑧𝑣 ↔ [𝑦 / 𝑥]𝜑))
105, 9anbi12i 614 . . . . 5 (([𝑦 / 𝑥]𝑣𝐴 ∧ [𝑦 / 𝑥]∀𝑧(𝑧𝑣𝜑)) ↔ (𝑣𝐴 ∧ ∀𝑧(𝑧𝑣 ↔ [𝑦 / 𝑥]𝜑)))
112, 10bitri 266 . . . 4 ([𝑦 / 𝑥](𝑣𝐴 ∧ ∀𝑧(𝑧𝑣𝜑)) ↔ (𝑣𝐴 ∧ ∀𝑧(𝑧𝑣 ↔ [𝑦 / 𝑥]𝜑)))
1211exbii 1933 . . 3 (∃𝑣[𝑦 / 𝑥](𝑣𝐴 ∧ ∀𝑧(𝑧𝑣𝜑)) ↔ ∃𝑣(𝑣𝐴 ∧ ∀𝑧(𝑧𝑣 ↔ [𝑦 / 𝑥]𝜑)))
131, 12bitri 266 . 2 ([𝑦 / 𝑥]∃𝑣(𝑣𝐴 ∧ ∀𝑧(𝑧𝑣𝜑)) ↔ ∃𝑣(𝑣𝐴 ∧ ∀𝑧(𝑧𝑣 ↔ [𝑦 / 𝑥]𝜑)))
14 clabel 2933 . . 3 ({𝑧𝜑} ∈ 𝐴 ↔ ∃𝑣(𝑣𝐴 ∧ ∀𝑧(𝑧𝑣𝜑)))
1514sbbii 2067 . 2 ([𝑦 / 𝑥]{𝑧𝜑} ∈ 𝐴 ↔ [𝑦 / 𝑥]∃𝑣(𝑣𝐴 ∧ ∀𝑧(𝑧𝑣𝜑)))
16 clabel 2933 . 2 ({𝑧 ∣ [𝑦 / 𝑥]𝜑} ∈ 𝐴 ↔ ∃𝑣(𝑣𝐴 ∧ ∀𝑧(𝑧𝑣 ↔ [𝑦 / 𝑥]𝜑)))
1713, 15, 163bitr4i 294 1 ([𝑦 / 𝑥]{𝑧𝜑} ∈ 𝐴 ↔ {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384  wal 1635  wex 1859  [wsb 2060  wcel 2156  {cab 2792  wnfc 2935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937
This theorem is referenced by: (None)
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