Theorem bnj1133 34625

Description: Technical lemma for bnj69 34646. 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
bnj1133.3	⊢ 𝐷 = (ω ∖ {∅})
bnj1133.5	⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
bnj1133.7	⊢ (𝜏 ↔ ∀𝑗 ∈ 𝑛 (𝑗 E 𝑖 → [𝑗 / 𝑖]𝜃))
bnj1133.8	⊢ ((𝑖 ∈ 𝑛 ∧ 𝜏) → 𝜃)

Assertion

Ref	Expression
bnj1133	⊢ (𝜒 → ∀𝑖 ∈ 𝑛 𝜃)

Distinct variable groups:   𝑖,𝑗,𝑛   𝜃,𝑗

Allowed substitution hints:   𝜑(𝑓,𝑖,𝑗,𝑛)   𝜓(𝑓,𝑖,𝑗,𝑛)   𝜒(𝑓,𝑖,𝑗,𝑛)   𝜃(𝑓,𝑖,𝑛)   𝜏(𝑓,𝑖,𝑗,𝑛)   𝐷(𝑓,𝑖,𝑗,𝑛)

Proof of Theorem bnj1133

Step	Hyp	Ref	Expression
1		bnj1133.5	. . 3 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
2		bnj1133.3	. . . 4 ⊢ 𝐷 = (ω ∖ {∅})
3	2	bnj1071 34613	. . 3 ⊢ (𝑛 ∈ 𝐷 → E Fr 𝑛)
4	1, 3	bnj769 34398	. 2 ⊢ (𝜒 → E Fr 𝑛)
5		impexp 449	. . . . . 6 ⊢ (((𝑖 ∈ 𝑛 ∧ 𝜏) → 𝜃) ↔ (𝑖 ∈ 𝑛 → (𝜏 → 𝜃)))
6	5	bicomi 223	. . . . 5 ⊢ ((𝑖 ∈ 𝑛 → (𝜏 → 𝜃)) ↔ ((𝑖 ∈ 𝑛 ∧ 𝜏) → 𝜃))
7	6	albii 1813	. . . 4 ⊢ (∀𝑖(𝑖 ∈ 𝑛 → (𝜏 → 𝜃)) ↔ ∀𝑖((𝑖 ∈ 𝑛 ∧ 𝜏) → 𝜃))
8		bnj1133.8	. . . 4 ⊢ ((𝑖 ∈ 𝑛 ∧ 𝜏) → 𝜃)
9	7, 8	mpgbir 1793	. . 3 ⊢ ∀𝑖(𝑖 ∈ 𝑛 → (𝜏 → 𝜃))
10		df-ral 3058	. . 3 ⊢ (∀𝑖 ∈ 𝑛 (𝜏 → 𝜃) ↔ ∀𝑖(𝑖 ∈ 𝑛 → (𝜏 → 𝜃)))
11	9, 10	mpbir 230	. 2 ⊢ ∀𝑖 ∈ 𝑛 (𝜏 → 𝜃)
12		vex 3475	. . 3 ⊢ 𝑛 ∈ V
13		bnj1133.7	. . 3 ⊢ (𝜏 ↔ ∀𝑗 ∈ 𝑛 (𝑗 E 𝑖 → [𝑗 / 𝑖]𝜃))
14	12, 13	bnj110 34494	. 2 ⊢ (( E Fr 𝑛 ∧ ∀𝑖 ∈ 𝑛 (𝜏 → 𝜃)) → ∀𝑖 ∈ 𝑛 𝜃)
15	4, 11, 14	sylancl 584	1 ⊢ (𝜒 → ∀𝑖 ∈ 𝑛 𝜃)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394  ∀wal 1531   = wceq 1533   ∈ wcel 2098  ∀wral 3057  [wsbc 3776   ∖ cdif 3944  ∅c0 4324  {csn 4630   class class class wbr 5150   E cep 5583   Fr wfr 5632   Fn wfn 6546  ωcom 7874   ∧ w-bnj17 34322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-tr 5268  df-po 5592  df-so 5593  df-fr 5635  df-we 5637  df-ord 6375  df-on 6376  df-om 7875  df-bnj17 34323

This theorem is referenced by:  bnj1128  34626