Theorem bnj121 34876

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj121.1	⊢ (𝜁 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
bnj121.2	⊢ (𝜁′ ↔ [1o / 𝑛]𝜁)
bnj121.3	⊢ (𝜑′ ↔ [1o / 𝑛]𝜑)
bnj121.4	⊢ (𝜓′ ↔ [1o / 𝑛]𝜓)

Assertion

Ref	Expression
bnj121	⊢ (𝜁′ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)))

Distinct variable groups:   𝐴,𝑛   𝑅,𝑛   𝑓,𝑛   𝑥,𝑛

Allowed substitution hints:   𝜑(𝑥,𝑓,𝑛)   𝜓(𝑥,𝑓,𝑛)   𝜁(𝑥,𝑓,𝑛)   𝐴(𝑥,𝑓)   𝑅(𝑥,𝑓)   𝜑′(𝑥,𝑓,𝑛)   𝜓′(𝑥,𝑓,𝑛)   𝜁′(𝑥,𝑓,𝑛)

Proof of Theorem bnj121

Step	Hyp	Ref	Expression
1		bnj121.1	. . 3 ⊢ (𝜁 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
2	1	sbcbii 3853	. 2 ⊢ ([1o / 𝑛]𝜁 ↔ [1o / 𝑛]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
3		bnj121.2	. 2 ⊢ (𝜁′ ↔ [1o / 𝑛]𝜁)
4		bnj105 34730	. . . . . . . 8 ⊢ 1o ∈ V
5	4	bnj90 34728	. . . . . . 7 ⊢ ([1o / 𝑛]𝑓 Fn 𝑛 ↔ 𝑓 Fn 1o)
6	5	bicomi 224	. . . . . 6 ⊢ (𝑓 Fn 1o ↔ [1o / 𝑛]𝑓 Fn 𝑛)
7		bnj121.3	. . . . . 6 ⊢ (𝜑′ ↔ [1o / 𝑛]𝜑)
8		bnj121.4	. . . . . 6 ⊢ (𝜓′ ↔ [1o / 𝑛]𝜓)
9	6, 7, 8	3anbi123i 1155	. . . . 5 ⊢ ((𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′) ↔ ([1o / 𝑛]𝑓 Fn 𝑛 ∧ [1o / 𝑛]𝜑 ∧ [1o / 𝑛]𝜓))
10		sbc3an 3862	. . . . 5 ⊢ ([1o / 𝑛](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ([1o / 𝑛]𝑓 Fn 𝑛 ∧ [1o / 𝑛]𝜑 ∧ [1o / 𝑛]𝜓))
11	9, 10	bitr4i 278	. . . 4 ⊢ ((𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′) ↔ [1o / 𝑛](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
12	11	imbi2i 336	. . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → [1o / 𝑛](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
13		nfv 1913	. . . . 5 ⊢ Ⅎ𝑛(𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴)
14	13	sbc19.21g 3870	. . . 4 ⊢ (1o ∈ V → ([1o / 𝑛]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → [1o / 𝑛](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))))
15	4, 14	ax-mp 5	. . 3 ⊢ ([1o / 𝑛]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → [1o / 𝑛](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
16	12, 15	bitr4i 278	. 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) ↔ [1o / 𝑛]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
17	2, 3, 16	3bitr4i 303	1 ⊢ (𝜁′ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2107  Vcvv 3479  [wsbc 3792   Fn wfn 6561  1_oc1o 8504   FrSe w-bnj15 34698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2707  ax-sep 5303  ax-nul 5313  ax-pow 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1541  df-fal 1551  df-ex 1778  df-nf 1782  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-sbc 3793  df-dif 3967  df-un 3969  df-ss 3981  df-nul 4341  df-pw 4608  df-sn 4633  df-suc 6395  df-fn 6569  df-1o 8511

This theorem is referenced by:  bnj150  34882  bnj153  34886