MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbabel Structured version   Visualization version   GIF version

Theorem sbabel 2778
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 2450 . . 3 ([𝑦 / 𝑥]∃𝑣(𝑣𝐴 ∧ ∀𝑧(𝑧𝑣𝜑)) ↔ ∃𝑣[𝑦 / 𝑥](𝑣𝐴 ∧ ∀𝑧(𝑧𝑣𝜑)))
2 sban 2386 . . . . 5 ([𝑦 / 𝑥](𝑣𝐴 ∧ ∀𝑧(𝑧𝑣𝜑)) ↔ ([𝑦 / 𝑥]𝑣𝐴 ∧ [𝑦 / 𝑥]∀𝑧(𝑧𝑣𝜑)))
3 sbabel.1 . . . . . . . 8 𝑥𝐴
43nfcri 2744 . . . . . . 7 𝑥 𝑣𝐴
54sbf 2367 . . . . . 6 ([𝑦 / 𝑥]𝑣𝐴𝑣𝐴)
6 nfv 1829 . . . . . . . . 9 𝑥 𝑧𝑣
76sbf 2367 . . . . . . . 8 ([𝑦 / 𝑥]𝑧𝑣𝑧𝑣)
87sbrbis 2392 . . . . . . 7 ([𝑦 / 𝑥](𝑧𝑣𝜑) ↔ (𝑧𝑣 ↔ [𝑦 / 𝑥]𝜑))
98sbalv 2451 . . . . . 6 ([𝑦 / 𝑥]∀𝑧(𝑧𝑣𝜑) ↔ ∀𝑧(𝑧𝑣 ↔ [𝑦 / 𝑥]𝜑))
105, 9anbi12i 728 . . . . 5 (([𝑦 / 𝑥]𝑣𝐴 ∧ [𝑦 / 𝑥]∀𝑧(𝑧𝑣𝜑)) ↔ (𝑣𝐴 ∧ ∀𝑧(𝑧𝑣 ↔ [𝑦 / 𝑥]𝜑)))
112, 10bitri 262 . . . 4 ([𝑦 / 𝑥](𝑣𝐴 ∧ ∀𝑧(𝑧𝑣𝜑)) ↔ (𝑣𝐴 ∧ ∀𝑧(𝑧𝑣 ↔ [𝑦 / 𝑥]𝜑)))
1211exbii 1763 . . 3 (∃𝑣[𝑦 / 𝑥](𝑣𝐴 ∧ ∀𝑧(𝑧𝑣𝜑)) ↔ ∃𝑣(𝑣𝐴 ∧ ∀𝑧(𝑧𝑣 ↔ [𝑦 / 𝑥]𝜑)))
131, 12bitri 262 . 2 ([𝑦 / 𝑥]∃𝑣(𝑣𝐴 ∧ ∀𝑧(𝑧𝑣𝜑)) ↔ ∃𝑣(𝑣𝐴 ∧ ∀𝑧(𝑧𝑣 ↔ [𝑦 / 𝑥]𝜑)))
14 clabel 2735 . . 3 ({𝑧𝜑} ∈ 𝐴 ↔ ∃𝑣(𝑣𝐴 ∧ ∀𝑧(𝑧𝑣𝜑)))
1514sbbii 1873 . 2 ([𝑦 / 𝑥]{𝑧𝜑} ∈ 𝐴 ↔ [𝑦 / 𝑥]∃𝑣(𝑣𝐴 ∧ ∀𝑧(𝑧𝑣𝜑)))
16 clabel 2735 . 2 ({𝑧 ∣ [𝑦 / 𝑥]𝜑} ∈ 𝐴 ↔ ∃𝑣(𝑣𝐴 ∧ ∀𝑧(𝑧𝑣 ↔ [𝑦 / 𝑥]𝜑)))
1713, 15, 163bitr4i 290 1 ([𝑦 / 𝑥]{𝑧𝜑} ∈ 𝐴 ↔ {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382  wal 1472  wex 1694  [wsb 1866  wcel 1976  {cab 2595  wnfc 2737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator