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Theorem bnj207 32039
Description: Technical lemma for bnj852 32079. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj207.1 (𝜒 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
bnj207.2 (𝜑′[𝑀 / 𝑛]𝜑)
bnj207.3 (𝜓′[𝑀 / 𝑛]𝜓)
bnj207.4 (𝜒′[𝑀 / 𝑛]𝜒)
bnj207.5 𝑀 ∈ V
Assertion
Ref Expression
bnj207 (𝜒′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑀𝜑′𝜓′)))
Distinct variable groups:   𝐴,𝑛   𝑓,𝑀   𝑅,𝑛   𝑓,𝑛   𝑥,𝑛
Allowed substitution hints:   𝜑(𝑥,𝑓,𝑛)   𝜓(𝑥,𝑓,𝑛)   𝜒(𝑥,𝑓,𝑛)   𝐴(𝑥,𝑓)   𝑅(𝑥,𝑓)   𝑀(𝑥,𝑛)   𝜑′(𝑥,𝑓,𝑛)   𝜓′(𝑥,𝑓,𝑛)   𝜒′(𝑥,𝑓,𝑛)

Proof of Theorem bnj207
StepHypRef Expression
1 bnj207.4 . 2 (𝜒′[𝑀 / 𝑛]𝜒)
2 bnj207.1 . . . 4 (𝜒 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
32sbcbii 3833 . . 3 ([𝑀 / 𝑛]𝜒[𝑀 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
4 bnj207.5 . . . . 5 𝑀 ∈ V
5 nfv 1908 . . . . . 6 𝑛(𝑅 FrSe 𝐴𝑥𝐴)
65sbc19.21g 3850 . . . . 5 (𝑀 ∈ V → ([𝑀 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → [𝑀 / 𝑛]∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))))
74, 6ax-mp 5 . . . 4 ([𝑀 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → [𝑀 / 𝑛]∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
84bnj89 31877 . . . . . 6 ([𝑀 / 𝑛]∃!𝑓(𝑓 Fn 𝑛𝜑𝜓) ↔ ∃!𝑓[𝑀 / 𝑛](𝑓 Fn 𝑛𝜑𝜓))
94bnj90 31878 . . . . . . . . 9 ([𝑀 / 𝑛]𝑓 Fn 𝑛𝑓 Fn 𝑀)
109bicomi 225 . . . . . . . 8 (𝑓 Fn 𝑀[𝑀 / 𝑛]𝑓 Fn 𝑛)
11 bnj207.2 . . . . . . . 8 (𝜑′[𝑀 / 𝑛]𝜑)
12 bnj207.3 . . . . . . . 8 (𝜓′[𝑀 / 𝑛]𝜓)
1310, 11, 12, 4bnj206 31887 . . . . . . 7 ([𝑀 / 𝑛](𝑓 Fn 𝑛𝜑𝜓) ↔ (𝑓 Fn 𝑀𝜑′𝜓′))
1413eubii 2668 . . . . . 6 (∃!𝑓[𝑀 / 𝑛](𝑓 Fn 𝑛𝜑𝜓) ↔ ∃!𝑓(𝑓 Fn 𝑀𝜑′𝜓′))
158, 14bitri 276 . . . . 5 ([𝑀 / 𝑛]∃!𝑓(𝑓 Fn 𝑛𝜑𝜓) ↔ ∃!𝑓(𝑓 Fn 𝑀𝜑′𝜓′))
1615imbi2i 337 . . . 4 (((𝑅 FrSe 𝐴𝑥𝐴) → [𝑀 / 𝑛]∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑀𝜑′𝜓′)))
177, 16bitri 276 . . 3 ([𝑀 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑀𝜑′𝜓′)))
183, 17bitri 276 . 2 ([𝑀 / 𝑛]𝜒 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑀𝜑′𝜓′)))
191, 18bitri 276 1 (𝜒′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑀𝜑′𝜓′)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081  wcel 2107  ∃!weu 2651  Vcvv 3500  [wsbc 3776   Fn wfn 6347   FrSe w-bnj15 31848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-v 3502  df-sbc 3777  df-fn 6355
This theorem is referenced by:  bnj600  32077  bnj908  32089
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