Theorem bnj985 31540

Description: Technical lemma for bnj69 31595. 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
bnj985.3	⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
bnj985.6	⊢ (𝜒′ ↔ [𝑝 / 𝑛]𝜒)
bnj985.9	⊢ (𝜒″ ↔ [𝐺 / 𝑓]𝜒′)
bnj985.11	⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
bnj985.13	⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})

Assertion

Ref	Expression
bnj985	⊢ (𝐺 ∈ 𝐵 ↔ ∃𝑝𝜒″)

Distinct variable groups:   𝐺,𝑝   𝜒,𝑝   𝑓,𝑝

Allowed substitution hints:   𝜑(𝑓,𝑛,𝑝)   𝜓(𝑓,𝑛,𝑝)   𝜒(𝑓,𝑛)   𝐵(𝑓,𝑛,𝑝)   𝐶(𝑓,𝑛,𝑝)   𝐷(𝑓,𝑛,𝑝)   𝐺(𝑓,𝑛)   𝜒′(𝑓,𝑛,𝑝)   𝜒″(𝑓,𝑛,𝑝)

Proof of Theorem bnj985

Step	Hyp	Ref	Expression
1		bnj985.13	. . . 4 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
2	1	bnj918 31353	. . 3 ⊢ 𝐺 ∈ V
3		bnj985.3	. . . 4 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
4		bnj985.11	. . . 4 ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
5	3, 4	bnj984 31539	. . 3 ⊢ (𝐺 ∈ V → (𝐺 ∈ 𝐵 ↔ [𝐺 / 𝑓]∃𝑛𝜒))
6	2, 5	ax-mp 5	. 2 ⊢ (𝐺 ∈ 𝐵 ↔ [𝐺 / 𝑓]∃𝑛𝜒)
7		sbcex2 3684	. . 3 ⊢ ([𝐺 / 𝑓]∃𝑝𝜒′ ↔ ∃𝑝[𝐺 / 𝑓]𝜒′)
8		nfv 2010	. . . . . . 7 ⊢ Ⅎ𝑝𝜒
9	8	sb8e 2543	. . . . . 6 ⊢ (∃𝑛𝜒 ↔ ∃𝑝[𝑝 / 𝑛]𝜒)
10		sbsbc 3637	. . . . . . 7 ⊢ ([𝑝 / 𝑛]𝜒 ↔ [𝑝 / 𝑛]𝜒)
11	10	exbii 1944	. . . . . 6 ⊢ (∃𝑝[𝑝 / 𝑛]𝜒 ↔ ∃𝑝[𝑝 / 𝑛]𝜒)
12	9, 11	bitri 267	. . . . 5 ⊢ (∃𝑛𝜒 ↔ ∃𝑝[𝑝 / 𝑛]𝜒)
13		bnj985.6	. . . . 5 ⊢ (𝜒′ ↔ [𝑝 / 𝑛]𝜒)
14	12, 13	bnj133 31313	. . . 4 ⊢ (∃𝑛𝜒 ↔ ∃𝑝𝜒′)
15	14	sbcbii 3689	. . 3 ⊢ ([𝐺 / 𝑓]∃𝑛𝜒 ↔ [𝐺 / 𝑓]∃𝑝𝜒′)
16		bnj985.9	. . . 4 ⊢ (𝜒″ ↔ [𝐺 / 𝑓]𝜒′)
17	16	exbii 1944	. . 3 ⊢ (∃𝑝𝜒″ ↔ ∃𝑝[𝐺 / 𝑓]𝜒′)
18	7, 15, 17	3bitr4i 295	. 2 ⊢ ([𝐺 / 𝑓]∃𝑛𝜒 ↔ ∃𝑝𝜒″)
19	6, 18	bitri 267	1 ⊢ (𝐺 ∈ 𝐵 ↔ ∃𝑝𝜒″)

Syntax hints:   ↔ wb 198   ∧ w3a 1108   = wceq 1653  ∃wex 1875  [wsb 2064   ∈ wcel 2157  {cab 2785  ∃wrex 3090  Vcvv 3385  [wsbc 3633   ∪ cun 3767  {csn 4368  ⟨cop 4374   Fn wfn 6096   ∧ w-bnj17 31272

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-8 2159  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-pr 5097  ax-un 7183

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-rex 3095  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-nul 4116  df-sn 4369  df-pr 4371  df-uni 4629  df-bnj17 31273

This theorem is referenced by:  bnj1018  31549