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Theorem bnj89 30915
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj89.1 𝑍 ∈ V
Assertion
Ref Expression
bnj89 ([𝑍 / 𝑦]∃!𝑥𝜑 ↔ ∃!𝑥[𝑍 / 𝑦]𝜑)
Distinct variable groups:   𝑥,𝑍   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑍(𝑦)

Proof of Theorem bnj89
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 sbcex2 3519 . . 3 ([𝑍 / 𝑦]𝑤𝑥(𝜑𝑥 = 𝑤) ↔ ∃𝑤[𝑍 / 𝑦]𝑥(𝜑𝑥 = 𝑤))
2 sbcal 3518 . . . 4 ([𝑍 / 𝑦]𝑥(𝜑𝑥 = 𝑤) ↔ ∀𝑥[𝑍 / 𝑦](𝜑𝑥 = 𝑤))
32exbii 1814 . . 3 (∃𝑤[𝑍 / 𝑦]𝑥(𝜑𝑥 = 𝑤) ↔ ∃𝑤𝑥[𝑍 / 𝑦](𝜑𝑥 = 𝑤))
4 bnj89.1 . . . . . . 7 𝑍 ∈ V
5 sbcbig 3513 . . . . . . 7 (𝑍 ∈ V → ([𝑍 / 𝑦](𝜑𝑥 = 𝑤) ↔ ([𝑍 / 𝑦]𝜑[𝑍 / 𝑦]𝑥 = 𝑤)))
64, 5ax-mp 5 . . . . . 6 ([𝑍 / 𝑦](𝜑𝑥 = 𝑤) ↔ ([𝑍 / 𝑦]𝜑[𝑍 / 𝑦]𝑥 = 𝑤))
7 sbcg 3536 . . . . . . . 8 (𝑍 ∈ V → ([𝑍 / 𝑦]𝑥 = 𝑤𝑥 = 𝑤))
84, 7ax-mp 5 . . . . . . 7 ([𝑍 / 𝑦]𝑥 = 𝑤𝑥 = 𝑤)
98bibi2i 326 . . . . . 6 (([𝑍 / 𝑦]𝜑[𝑍 / 𝑦]𝑥 = 𝑤) ↔ ([𝑍 / 𝑦]𝜑𝑥 = 𝑤))
106, 9bitri 264 . . . . 5 ([𝑍 / 𝑦](𝜑𝑥 = 𝑤) ↔ ([𝑍 / 𝑦]𝜑𝑥 = 𝑤))
1110albii 1787 . . . 4 (∀𝑥[𝑍 / 𝑦](𝜑𝑥 = 𝑤) ↔ ∀𝑥([𝑍 / 𝑦]𝜑𝑥 = 𝑤))
1211exbii 1814 . . 3 (∃𝑤𝑥[𝑍 / 𝑦](𝜑𝑥 = 𝑤) ↔ ∃𝑤𝑥([𝑍 / 𝑦]𝜑𝑥 = 𝑤))
131, 3, 123bitri 286 . 2 ([𝑍 / 𝑦]𝑤𝑥(𝜑𝑥 = 𝑤) ↔ ∃𝑤𝑥([𝑍 / 𝑦]𝜑𝑥 = 𝑤))
14 df-eu 2502 . . 3 (∃!𝑥𝜑 ↔ ∃𝑤𝑥(𝜑𝑥 = 𝑤))
1514sbcbii 3524 . 2 ([𝑍 / 𝑦]∃!𝑥𝜑[𝑍 / 𝑦]𝑤𝑥(𝜑𝑥 = 𝑤))
16 df-eu 2502 . 2 (∃!𝑥[𝑍 / 𝑦]𝜑 ↔ ∃𝑤𝑥([𝑍 / 𝑦]𝜑𝑥 = 𝑤))
1713, 15, 163bitr4i 292 1 ([𝑍 / 𝑦]∃!𝑥𝜑 ↔ ∃!𝑥[𝑍 / 𝑦]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wal 1521  wex 1744  wcel 2030  ∃!weu 2498  Vcvv 3231  [wsbc 3468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-v 3233  df-sbc 3469
This theorem is referenced by:  bnj130  31070  bnj207  31077
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