Theorem bnj207 30205

Description: Technical lemma for bnj852 30245. 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 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
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 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑀 ∧ 𝜑′ ∧ 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

Step	Hyp	Ref	Expression
1		bnj207.4	. 2 ⊢ (𝜒′ ↔ [𝑀 / 𝑛]𝜒)
2		bnj207.1	. . . 4 ⊢ (𝜒 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
3	2	sbcbii 3458	. . 3 ⊢ ([𝑀 / 𝑛]𝜒 ↔ [𝑀 / 𝑛]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
4		bnj207.5	. . . . 5 ⊢ 𝑀 ∈ V
5		nfv 1830	. . . . . 6 ⊢ Ⅎ𝑛(𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴)
6	5	sbc19.21g 3469	. . . . 5 ⊢ (𝑀 ∈ V → ([𝑀 / 𝑛]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → [𝑀 / 𝑛]∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))))
7	4, 6	ax-mp 5	. . . 4 ⊢ ([𝑀 / 𝑛]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → [𝑀 / 𝑛]∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
8	4	bnj89 30041	. . . . . 6 ⊢ ([𝑀 / 𝑛]∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ∃!𝑓[𝑀 / 𝑛](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
9	4	bnj90 30042	. . . . . . . . 9 ⊢ ([𝑀 / 𝑛]𝑓 Fn 𝑛 ↔ 𝑓 Fn 𝑀)
10	9	bicomi 213	. . . . . . . 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)
13	10, 11, 12, 4	bnj206 30053	. . . . . . 7 ⊢ ([𝑀 / 𝑛](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (𝑓 Fn 𝑀 ∧ 𝜑′ ∧ 𝜓′))
14	13	eubii 2480	. . . . . 6 ⊢ (∃!𝑓[𝑀 / 𝑛](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ∃!𝑓(𝑓 Fn 𝑀 ∧ 𝜑′ ∧ 𝜓′))
15	8, 14	bitri 263	. . . . 5 ⊢ ([𝑀 / 𝑛]∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ∃!𝑓(𝑓 Fn 𝑀 ∧ 𝜑′ ∧ 𝜓′))
16	15	imbi2i 325	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → [𝑀 / 𝑛]∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑀 ∧ 𝜑′ ∧ 𝜓′)))
17	7, 16	bitri 263	. . 3 ⊢ ([𝑀 / 𝑛]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑀 ∧ 𝜑′ ∧ 𝜓′)))
18	3, 17	bitri 263	. 2 ⊢ ([𝑀 / 𝑛]𝜒 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑀 ∧ 𝜑′ ∧ 𝜓′)))
19	1, 18	bitri 263	1 ⊢ (𝜒′ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑀 ∧ 𝜑′ ∧ 𝜓′)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   ∈ wcel 1977  ∃!weu 2458  Vcvv 3173  [wsbc 3402   Fn wfn 5799   FrSe w-bnj15 30011

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175  df-sbc 3403  df-fn 5807

This theorem is referenced by:  bnj600  30243  bnj908  30255