Theorem bnj124 34346

Description: Technical lemma for bnj150 34351. 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 3837	. . 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 34339	. . . 4 ⊢ 𝐹 ∈ V
6		nfv 1916	. . . . 5 ⊢ Ⅎ𝑓(𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴)
7	6	sbc19.21g 3855	. . . 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 6640	. . . . . . . 8 ⊢ (𝑓 = 𝑧 → (𝑓 Fn 1o ↔ 𝑧 Fn 1o))
10		fneq1 6640	. . . . . . . 8 ⊢ (𝑧 = 𝐹 → (𝑧 Fn 1o ↔ 𝐹 Fn 1o))
11	9, 10	sbcie2g 3819	. . . . . . 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 34206	. . . 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 275	1 ⊢ (𝜁″ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹 Fn 1o ∧ 𝜑″ ∧ 𝜓″)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  Vcvv 3473  [wsbc 3777  ∅c0 4322  {csn 4628  ⟨cop 4634   Fn wfn 6538  1_oc1o 8465   predc-bnj14 34163   FrSe w-bnj15 34167

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-fun 6545  df-fn 6546

This theorem is referenced by:  bnj150  34351  bnj153  34355