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Theorem bnj89 32116
 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 3781 . . 3 ([𝑍 / 𝑦]𝑤𝑥(𝜑𝑥 = 𝑤) ↔ ∃𝑤[𝑍 / 𝑦]𝑥(𝜑𝑥 = 𝑤))
2 sbcal 3780 . . . 4 ([𝑍 / 𝑦]𝑥(𝜑𝑥 = 𝑤) ↔ ∀𝑥[𝑍 / 𝑦](𝜑𝑥 = 𝑤))
32exbii 1849 . . 3 (∃𝑤[𝑍 / 𝑦]𝑥(𝜑𝑥 = 𝑤) ↔ ∃𝑤𝑥[𝑍 / 𝑦](𝜑𝑥 = 𝑤))
4 bnj89.1 . . . . . . 7 𝑍 ∈ V
5 sbcbig 3770 . . . . . . 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 3793 . . . . . . . 8 (𝑍 ∈ V → ([𝑍 / 𝑦]𝑥 = 𝑤𝑥 = 𝑤))
84, 7ax-mp 5 . . . . . . 7 ([𝑍 / 𝑦]𝑥 = 𝑤𝑥 = 𝑤)
98bibi2i 341 . . . . . 6 (([𝑍 / 𝑦]𝜑[𝑍 / 𝑦]𝑥 = 𝑤) ↔ ([𝑍 / 𝑦]𝜑𝑥 = 𝑤))
106, 9bitri 278 . . . . 5 ([𝑍 / 𝑦](𝜑𝑥 = 𝑤) ↔ ([𝑍 / 𝑦]𝜑𝑥 = 𝑤))
1110albii 1821 . . . 4 (∀𝑥[𝑍 / 𝑦](𝜑𝑥 = 𝑤) ↔ ∀𝑥([𝑍 / 𝑦]𝜑𝑥 = 𝑤))
1211exbii 1849 . . 3 (∃𝑤𝑥[𝑍 / 𝑦](𝜑𝑥 = 𝑤) ↔ ∃𝑤𝑥([𝑍 / 𝑦]𝜑𝑥 = 𝑤))
131, 3, 123bitri 300 . 2 ([𝑍 / 𝑦]𝑤𝑥(𝜑𝑥 = 𝑤) ↔ ∃𝑤𝑥([𝑍 / 𝑦]𝜑𝑥 = 𝑤))
14 eu6 2634 . . 3 (∃!𝑥𝜑 ↔ ∃𝑤𝑥(𝜑𝑥 = 𝑤))
1514sbcbii 3776 . 2 ([𝑍 / 𝑦]∃!𝑥𝜑[𝑍 / 𝑦]𝑤𝑥(𝜑𝑥 = 𝑤))
16 eu6 2634 . 2 (∃!𝑥[𝑍 / 𝑦]𝜑 ↔ ∃𝑤𝑥([𝑍 / 𝑦]𝜑𝑥 = 𝑤))
1713, 15, 163bitr4i 306 1 ([𝑍 / 𝑦]∃!𝑥𝜑 ↔ ∃!𝑥[𝑍 / 𝑦]𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209  ∀wal 1536  ∃wex 1781   ∈ wcel 2111  ∃!weu 2628  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-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-sbc 3721 This theorem is referenced by:  bnj130  32271  bnj207  32278
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