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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1039 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 35002. 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.) |
Ref | Expression |
---|---|
bnj1039.1 | ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) |
bnj1039.2 | ⊢ (𝜓′ ↔ [𝑗 / 𝑖]𝜓) |
Ref | Expression |
---|---|
bnj1039 | ⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1039.2 | . 2 ⊢ (𝜓′ ↔ [𝑗 / 𝑖]𝜓) | |
2 | vex 3481 | . . 3 ⊢ 𝑗 ∈ V | |
3 | bnj1039.1 | . . . 4 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) | |
4 | nfra1 3281 | . . . 4 ⊢ Ⅎ𝑖∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) | |
5 | 3, 4 | nfxfr 1849 | . . 3 ⊢ Ⅎ𝑖𝜓 |
6 | 2, 5 | sbcgfi 3871 | . 2 ⊢ ([𝑗 / 𝑖]𝜓 ↔ 𝜓) |
7 | 1, 6, 3 | 3bitri 297 | 1 ⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1536 ∈ wcel 2105 ∀wral 3058 [wsbc 3790 ∪ ciun 4995 suc csuc 6387 ‘cfv 6562 ωcom 7886 predc-bnj14 34680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-12 2174 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1539 df-ex 1776 df-nf 1780 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-v 3479 df-sbc 3791 |
This theorem is referenced by: bnj1128 34982 |
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