Theorem bnj985 35450

Description: Technical lemma for bnj69 35506. 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 35449 for a version with more disjoint variable conditions, not requiring ax-13 2403. (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 35263	. . 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 35448	. . 3 ⊢ (𝐺 ∈ V → (𝐺 ∈ 𝐵 ↔ [𝐺 / 𝑓]∃𝑛𝜒))
6	2, 5	ax-mp 5	. 2 ⊢ (𝐺 ∈ 𝐵 ↔ [𝐺 / 𝑓]∃𝑛𝜒)
7		sbcex2 3802	. . 3 ⊢ ([𝐺 / 𝑓]∃𝑝𝜒′ ↔ ∃𝑝[𝐺 / 𝑓]𝜒′)
8		nfv 1947	. . . . . . 7 ⊢ Ⅎ𝑝𝜒
9	8	sb8e 2549	. . . . . 6 ⊢ (∃𝑛𝜒 ↔ ∃𝑝[𝑝 / 𝑛]𝜒)
10		sbsbc 3746	. . . . . . 7 ⊢ ([𝑝 / 𝑛]𝜒 ↔ [𝑝 / 𝑛]𝜒)
11	10	exbii 1881	. . . . . 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 35224	. . . 4 ⊢ (∃𝑛𝜒 ↔ ∃𝑝𝜒′)
15	14	sbcbii 3798	. . 3 ⊢ ([𝐺 / 𝑓]∃𝑛𝜒 ↔ [𝐺 / 𝑓]∃𝑝𝜒′)
16		bnj985.9	. . . 4 ⊢ (𝜒″ ↔ [𝐺 / 𝑓]𝜒′)
17	16	exbii 1881	. . 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″)

This proof depends on syntax axioms:   ↔ wb 209   ∧ w3a 1103   = wceq 1570  ∃wex 1812  [wsb 2099   ∈ wcel 2145  {cab 2740  ∃wrex 3088  Vcvv 3453  [wsbc 3742   ∪ cun 3900  {csn 4587  ⟨cop 4593   Fn wfn 6532   ∧ w-bnj17 35183

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-13 2403  ax-ext 2734  ax-sep 5255  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rex 3089  df-v 3455  df-sbc 3743  df-un 3907  df-ss 3919  df-sn 4588  df-pr 4590  df-uni 4871  df-bnj17 35184

This theorem is used by:  bnj1018g  35459