Theorem bnj121 35500

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 3794	. 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 35355	. . . . . . . 8 ⊢ 1o ∈ V
5	4	bnj90 35353	. . . . . . 7 ⊢ ([1o / 𝑛]𝑓 Fn 𝑛 ↔ 𝑓 Fn 1o)
6	5	bicomi 227	. . . . . 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 1173	. . . . 5 ⊢ ((𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′) ↔ ([1o / 𝑛]𝑓 Fn 𝑛 ∧ [1o / 𝑛]𝜑 ∧ [1o / 𝑛]𝜓))
10		sbc3an 3802	. . . . 5 ⊢ ([1o / 𝑛](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ([1o / 𝑛]𝑓 Fn 𝑛 ∧ [1o / 𝑛]𝜑 ∧ [1o / 𝑛]𝜓))
11	9, 10	bitr4i 281	. . . 4 ⊢ ((𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′) ↔ [1o / 𝑛](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
12	11	imbi2i 339	. . 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 1946	. . . . 5 ⊢ Ⅎ𝑛(𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴)
14	13	sbc19.21g 3809	. . . 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 281	. 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 306	1 ⊢ (𝜁′ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   ∈ wcel 2144  Vcvv 3450  [wsbc 3738   Fn wfn 6533  1_oc1o 8469   FrSe w-bnj15 35323

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-pw 4558  df-sn 4584  df-suc 6368  df-fn 6541  df-1o 8476

This theorem is used by:  bnj150  35506  bnj153  35510