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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj124 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj150 30946. 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 3491 | . . 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 30934 | . . . 4 ⊢ 𝐹 ∈ V |
6 | nfv 1843 | . . . . 5 ⊢ Ⅎ𝑓(𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) | |
7 | 6 | sbc19.21g 3502 | . . . 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 5979 | . . . . . . . 8 ⊢ (𝑓 = 𝑧 → (𝑓 Fn 1𝑜 ↔ 𝑧 Fn 1𝑜)) | |
10 | fneq1 5979 | . . . . . . . 8 ⊢ (𝑧 = 𝐹 → (𝑧 Fn 1𝑜 ↔ 𝐹 Fn 1𝑜)) | |
11 | 9, 10 | sbcie2g 3469 | . . . . . . 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 214 | . . . . 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 30799 | . . . 4 ⊢ ([𝐹 / 𝑓](𝑓 Fn 1𝑜 ∧ 𝜑′ ∧ 𝜓′) ↔ (𝐹 Fn 1𝑜 ∧ 𝜑″ ∧ 𝜓″)) |
17 | 16 | imbi2i 326 | . . 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 286 | . 2 ⊢ ([𝐹 / 𝑓]𝜁′ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹 Fn 1𝑜 ∧ 𝜑″ ∧ 𝜓″))) |
19 | 1, 18 | bitri 264 | 1 ⊢ (𝜁″ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹 Fn 1𝑜 ∧ 𝜑″ ∧ 𝜓″))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 Vcvv 3200 [wsbc 3435 ∅c0 3915 {csn 4177 〈cop 4183 Fn wfn 5883 1𝑜c1o 7553 predc-bnj14 30754 FrSe w-bnj15 30758 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-fun 5890 df-fn 5891 |
This theorem is referenced by: bnj150 30946 bnj153 30950 |
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