Theorem bnj121 31457

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 3689	. 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 31310	. . . . . . . 8 ⊢ 1𝑜 ∈ V
5	4	bnj90 31308	. . . . . . 7 ⊢ ([1𝑜 / 𝑛]𝑓 Fn 𝑛 ↔ 𝑓 Fn 1𝑜)
6	5	bicomi 216	. . . . . 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 1195	. . . . 5 ⊢ ((𝑓 Fn 1𝑜 ∧ 𝜑′ ∧ 𝜓′) ↔ ([1𝑜 / 𝑛]𝑓 Fn 𝑛 ∧ [1𝑜 / 𝑛]𝜑 ∧ [1𝑜 / 𝑛]𝜓))
10		sbc3an 3691	. . . . 5 ⊢ ([1𝑜 / 𝑛](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ([1𝑜 / 𝑛]𝑓 Fn 𝑛 ∧ [1𝑜 / 𝑛]𝜑 ∧ [1𝑜 / 𝑛]𝜓))
11	9, 10	bitr4i 270	. . . 4 ⊢ ((𝑓 Fn 1𝑜 ∧ 𝜑′ ∧ 𝜓′) ↔ [1𝑜 / 𝑛](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
12	11	imbi2i 328	. . 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 2010	. . . . 5 ⊢ Ⅎ𝑛(𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴)
14	13	sbc19.21g 3698	. . . 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 270	. 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 295	1 ⊢ (𝜁′ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 1𝑜 ∧ 𝜑′ ∧ 𝜓′)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 385   ∧ w3a 1108   ∈ wcel 2157  Vcvv 3385  [wsbc 3633   Fn wfn 6096  1_𝑜c1o 7792   FrSe w-bnj15 31278

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 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035

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-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-pw 4351  df-sn 4369  df-suc 5947  df-fn 6104  df-1o 7799

This theorem is referenced by:  bnj150  31463  bnj153  31467