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Theorem bnj207 31461
Description: Technical lemma for bnj852 31501. 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 3688 . . 3 ([𝑀 / 𝑛]𝜒[𝑀 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
4 bnj207.5 . . . . 5 𝑀 ∈ V
5 nfv 2010 . . . . . 6 𝑛(𝑅 FrSe 𝐴𝑥𝐴)
65sbc19.21g 3697 . . . . 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 31300 . . . . . 6 ([𝑀 / 𝑛]∃!𝑓(𝑓 Fn 𝑛𝜑𝜓) ↔ ∃!𝑓[𝑀 / 𝑛](𝑓 Fn 𝑛𝜑𝜓))
94bnj90 31301 . . . . . . . . 9 ([𝑀 / 𝑛]𝑓 Fn 𝑛𝑓 Fn 𝑀)
109bicomi 216 . . . . . . . 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 31310 . . . . . . 7 ([𝑀 / 𝑛](𝑓 Fn 𝑛𝜑𝜓) ↔ (𝑓 Fn 𝑀𝜑′𝜓′))
1413eubii 2625 . . . . . 6 (∃!𝑓[𝑀 / 𝑛](𝑓 Fn 𝑛𝜑𝜓) ↔ ∃!𝑓(𝑓 Fn 𝑀𝜑′𝜓′))
158, 14bitri 267 . . . . 5 ([𝑀 / 𝑛]∃!𝑓(𝑓 Fn 𝑛𝜑𝜓) ↔ ∃!𝑓(𝑓 Fn 𝑀𝜑′𝜓′))
1615imbi2i 328 . . . 4 (((𝑅 FrSe 𝐴𝑥𝐴) → [𝑀 / 𝑛]∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑀𝜑′𝜓′)))
177, 16bitri 267 . . 3 ([𝑀 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑀𝜑′𝜓′)))
183, 17bitri 267 . 2 ([𝑀 / 𝑛]𝜒 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑀𝜑′𝜓′)))
191, 18bitri 267 1 (𝜒′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑀𝜑′𝜓′)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108  wcel 2157  ∃!weu 2608  Vcvv 3384  [wsbc 3632   Fn wfn 6095   FrSe w-bnj15 31271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2776
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2785  df-cleq 2791  df-clel 2794  df-v 3386  df-sbc 3633  df-fn 6103
This theorem is referenced by:  bnj600  31499  bnj908  31511
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