Theorem bnj124 32851

Description: Technical lemma for bnj150 32856. 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.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj124.1	⊢ 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
bnj124.2	⊢ (𝜑″ ↔ [𝐹 / 𝑓]𝜑′)
bnj124.3	⊢ (𝜓″ ↔ [𝐹 / 𝑓]𝜓′)
bnj124.4	⊢ (𝜁″ ↔ [𝐹 / 𝑓]𝜁′)
bnj124.5	⊢ (𝜁′ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)))

Assertion

Ref	Expression
bnj124	⊢ (𝜁″ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹 Fn 1o ∧ 𝜑″ ∧ 𝜓″)))

Distinct variable groups:   𝐴,𝑓   𝑅,𝑓   𝑥,𝑓

Allowed substitution hints:   𝐴(𝑥)   𝑅(𝑥)   𝐹(𝑥,𝑓)   𝜑′(𝑥,𝑓)   𝜓′(𝑥,𝑓)   𝜁′(𝑥,𝑓)   𝜑″(𝑥,𝑓)   𝜓″(𝑥,𝑓)   𝜁″(𝑥,𝑓)

Proof of Theorem bnj124

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj124.4	. 2 ⊢ (𝜁″ ↔ [𝐹 / 𝑓]𝜁′)
2		bnj124.5	. . . 4 ⊢ (𝜁′ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)))
3	2	sbcbii 3776	. . 3 ⊢ ([𝐹 / 𝑓]𝜁′ ↔ [𝐹 / 𝑓]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)))
4		bnj124.1	. . . . 5 ⊢ 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
5	4	bnj95 32844	. . . 4 ⊢ 𝐹 ∈ V
6		nfv 1917	. . . . 5 ⊢ Ⅎ𝑓(𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴)
7	6	sbc19.21g 3794	. . . 4 ⊢ (𝐹 ∈ V → ([𝐹 / 𝑓]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → [𝐹 / 𝑓](𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′))))
8	5, 7	ax-mp 5	. . 3 ⊢ ([𝐹 / 𝑓]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → [𝐹 / 𝑓](𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)))
9		fneq1 6524	. . . . . . . 8 ⊢ (𝑓 = 𝑧 → (𝑓 Fn 1o ↔ 𝑧 Fn 1o))
10		fneq1 6524	. . . . . . . 8 ⊢ (𝑧 = 𝐹 → (𝑧 Fn 1o ↔ 𝐹 Fn 1o))
11	9, 10	sbcie2g 3758	. . . . . . 7 ⊢ (𝐹 ∈ V → ([𝐹 / 𝑓]𝑓 Fn 1o ↔ 𝐹 Fn 1o))
12	5, 11	ax-mp 5	. . . . . 6 ⊢ ([𝐹 / 𝑓]𝑓 Fn 1o ↔ 𝐹 Fn 1o)
13	12	bicomi 223	. . . . 5 ⊢ (𝐹 Fn 1o ↔ [𝐹 / 𝑓]𝑓 Fn 1o)
14		bnj124.2	. . . . 5 ⊢ (𝜑″ ↔ [𝐹 / 𝑓]𝜑′)
15		bnj124.3	. . . . 5 ⊢ (𝜓″ ↔ [𝐹 / 𝑓]𝜓′)
16	13, 14, 15, 5	bnj206 32710	. . . 4 ⊢ ([𝐹 / 𝑓](𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′) ↔ (𝐹 Fn 1o ∧ 𝜑″ ∧ 𝜓″))
17	16	imbi2i 336	. . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → [𝐹 / 𝑓](𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹 Fn 1o ∧ 𝜑″ ∧ 𝜓″)))
18	3, 8, 17	3bitri 297	. 2 ⊢ ([𝐹 / 𝑓]𝜁′ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹 Fn 1o ∧ 𝜑″ ∧ 𝜓″)))
19	1, 18	bitri 274	1 ⊢ (𝜁″ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹 Fn 1o ∧ 𝜑″ ∧ 𝜓″)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  Vcvv 3432  [wsbc 3716  ∅c0 4256  {csn 4561  ⟨cop 4567   Fn wfn 6428  1_oc1o 8290   predc-bnj14 32667   FrSe w-bnj15 32671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-fun 6435  df-fn 6436

This theorem is referenced by:  bnj150  32856  bnj153  32860