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| Description: Technical lemma for bnj150 34890. 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.) | 
| 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′))) | 
| Ref | Expression | 
|---|---|
| bnj124 | ⊢ (𝜁″ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹 Fn 1o ∧ 𝜑″ ∧ 𝜓″))) | 
| 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 3846 | . . 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 34878 | . . . 4 ⊢ 𝐹 ∈ V | 
| 6 | nfv 1914 | . . . . 5 ⊢ Ⅎ𝑓(𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) | |
| 7 | 6 | sbc19.21g 3862 | . . . 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 6659 | . . . . . . . 8 ⊢ (𝑓 = 𝑧 → (𝑓 Fn 1o ↔ 𝑧 Fn 1o)) | |
| 10 | fneq1 6659 | . . . . . . . 8 ⊢ (𝑧 = 𝐹 → (𝑧 Fn 1o ↔ 𝐹 Fn 1o)) | |
| 11 | 9, 10 | sbcie2g 3829 | . . . . . . 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 224 | . . . . 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 34745 | . . . 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″))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 Vcvv 3480 [wsbc 3788 ∅c0 4333 {csn 4626 〈cop 4632 Fn wfn 6556 1oc1o 8499 predc-bnj14 34702 FrSe w-bnj15 34706 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-fun 6563 df-fn 6564 | 
| This theorem is referenced by: bnj150 34890 bnj153 34894 | 
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