Theorem bnj984 34583

Description: Technical lemma for bnj69 34641. 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
bnj984.3	⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
bnj984.11	⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}

Assertion

Ref	Expression
bnj984	⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ 𝐵 ↔ [𝐺 / 𝑓]∃𝑛𝜒))

Proof of Theorem bnj984

Step	Hyp	Ref	Expression
1		bnj984.11	. . . 4 ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
2	1	eleq2i 2821	. . 3 ⊢ (𝐺 ∈ 𝐵 ↔ 𝐺 ∈ {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)})
3		sbc8g 3784	. . 3 ⊢ (𝐺 ∈ 𝐴 → ([𝐺 / 𝑓]∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ 𝐺 ∈ {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}))
4	2, 3	bitr4id 290	. 2 ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ 𝐵 ↔ [𝐺 / 𝑓]∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
5		df-rex 3068	. . . 4 ⊢ (∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ∃𝑛(𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
6		bnj984.3	. . . . 5 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
7		bnj252 34334	. . . . 5 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
8	6, 7	bitri 275	. . . 4 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
9	5, 8	bnj133 34358	. . 3 ⊢ (∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ∃𝑛𝜒)
10	9	sbcbii 3837	. 2 ⊢ ([𝐺 / 𝑓]∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ [𝐺 / 𝑓]∃𝑛𝜒)
11	4, 10	bitrdi 287	1 ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ 𝐵 ↔ [𝐺 / 𝑓]∃𝑛𝜒))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534  ∃wex 1774   ∈ wcel 2099  {cab 2705  ∃wrex 3067  [wsbc 3776   Fn wfn 6543   ∧ w-bnj17 34317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1087  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rex 3068  df-sbc 3777  df-bnj17 34318

This theorem is referenced by:  bnj985v  34584  bnj985  34585