Theorem bnj985 32336

Description: Technical lemma for bnj69 32392. 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). See bnj985v 32335 for a version with more disjoint variable conditions, not requiring ax-13 2379. (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 32147	. . 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 32334	. . 3 ⊢ (𝐺 ∈ V → (𝐺 ∈ 𝐵 ↔ [𝐺 / 𝑓]∃𝑛𝜒))
6	2, 5	ax-mp 5	. 2 ⊢ (𝐺 ∈ 𝐵 ↔ [𝐺 / 𝑓]∃𝑛𝜒)
7		sbcex2 3781	. . 3 ⊢ ([𝐺 / 𝑓]∃𝑝𝜒′ ↔ ∃𝑝[𝐺 / 𝑓]𝜒′)
8		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑝𝜒
9	8	sb8e 2537	. . . . . 6 ⊢ (∃𝑛𝜒 ↔ ∃𝑝[𝑝 / 𝑛]𝜒)
10		sbsbc 3724	. . . . . . 7 ⊢ ([𝑝 / 𝑛]𝜒 ↔ [𝑝 / 𝑛]𝜒)
11	10	exbii 1849	. . . . . 6 ⊢ (∃𝑝[𝑝 / 𝑛]𝜒 ↔ ∃𝑝[𝑝 / 𝑛]𝜒)
12	9, 11	bitri 278	. . . . 5 ⊢ (∃𝑛𝜒 ↔ ∃𝑝[𝑝 / 𝑛]𝜒)
13		bnj985.6	. . . . 5 ⊢ (𝜒′ ↔ [𝑝 / 𝑛]𝜒)
14	12, 13	bnj133 32107	. . . 4 ⊢ (∃𝑛𝜒 ↔ ∃𝑝𝜒′)
15	14	sbcbii 3776	. . 3 ⊢ ([𝐺 / 𝑓]∃𝑛𝜒 ↔ [𝐺 / 𝑓]∃𝑝𝜒′)
16		bnj985.9	. . . 4 ⊢ (𝜒″ ↔ [𝐺 / 𝑓]𝜒′)
17	16	exbii 1849	. . 3 ⊢ (∃𝑝𝜒″ ↔ ∃𝑝[𝐺 / 𝑓]𝜒′)
18	7, 15, 17	3bitr4i 306	. 2 ⊢ ([𝐺 / 𝑓]∃𝑛𝜒 ↔ ∃𝑝𝜒″)
19	6, 18	bitri 278	1 ⊢ (𝐺 ∈ 𝐵 ↔ ∃𝑝𝜒″)

Syntax hints:   ↔ wb 209   ∧ w3a 1084   = wceq 1538  ∃wex 1781  [wsb 2069   ∈ wcel 2111  {cab 2776  ∃wrex 3107  Vcvv 3441  [wsbc 3720   ∪ cun 3879  {csn 4525  ⟨cop 4531   Fn wfn 6319   ∧ w-bnj17 32066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-13 2379  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-rex 3112  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-sn 4526  df-pr 4528  df-uni 4801  df-bnj17 32067

This theorem is referenced by:  bnj1018g  32345