Theorem bnj985 35110

Description: Technical lemma for bnj69 35166. 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 35109 for a version with more disjoint variable conditions, not requiring ax-13 2376. (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 34922	. . 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 35108	. . 3 ⊢ (𝐺 ∈ V → (𝐺 ∈ 𝐵 ↔ [𝐺 / 𝑓]∃𝑛𝜒))
6	2, 5	ax-mp 5	. 2 ⊢ (𝐺 ∈ 𝐵 ↔ [𝐺 / 𝑓]∃𝑛𝜒)
7		sbcex2 3801	. . 3 ⊢ ([𝐺 / 𝑓]∃𝑝𝜒′ ↔ ∃𝑝[𝐺 / 𝑓]𝜒′)
8		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑝𝜒
9	8	sb8e 2522	. . . . . 6 ⊢ (∃𝑛𝜒 ↔ ∃𝑝[𝑝 / 𝑛]𝜒)
10		sbsbc 3744	. . . . . . 7 ⊢ ([𝑝 / 𝑛]𝜒 ↔ [𝑝 / 𝑛]𝜒)
11	10	exbii 1849	. . . . . 6 ⊢ (∃𝑝[𝑝 / 𝑛]𝜒 ↔ ∃𝑝[𝑝 / 𝑛]𝜒)
12	9, 11	bitri 275	. . . . 5 ⊢ (∃𝑛𝜒 ↔ ∃𝑝[𝑝 / 𝑛]𝜒)
13		bnj985.6	. . . . 5 ⊢ (𝜒′ ↔ [𝑝 / 𝑛]𝜒)
14	12, 13	bnj133 34883	. . . 4 ⊢ (∃𝑛𝜒 ↔ ∃𝑝𝜒′)
15	14	sbcbii 3797	. . 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 303	. 2 ⊢ ([𝐺 / 𝑓]∃𝑛𝜒 ↔ ∃𝑝𝜒″)
19	6, 18	bitri 275	1 ⊢ (𝐺 ∈ 𝐵 ↔ ∃𝑝𝜒″)

Syntax hints:   ↔ wb 206   ∧ w3a 1086   = wceq 1541  ∃wex 1780  [wsb 2067   ∈ wcel 2113  {cab 2714  ∃wrex 3060  Vcvv 3440  [wsbc 3740   ∪ cun 3899  {csn 4580  ⟨cop 4586   Fn wfn 6487   ∧ w-bnj17 34842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-13 2376  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rex 3061  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-sn 4581  df-pr 4583  df-uni 4864  df-bnj17 34843

This theorem is referenced by:  bnj1018g  35119