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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1039 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 32703. 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 3412 | . . 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 3140 | . . . 4 ⊢ Ⅎ𝑖∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) | |
5 | 3, 4 | nfxfr 1860 | . . 3 ⊢ Ⅎ𝑖𝜓 |
6 | 2, 5 | sbcgfi 3776 | . 2 ⊢ ([𝑗 / 𝑖]𝜓 ↔ 𝜓) |
7 | 1, 6, 3 | 3bitri 300 | 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 209 = wceq 1543 ∈ wcel 2110 ∀wral 3061 [wsbc 3694 ∪ ciun 4904 suc csuc 6215 ‘cfv 6380 ωcom 7644 predc-bnj14 32379 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-12 2175 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-nf 1792 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3066 df-v 3410 df-sbc 3695 |
This theorem is referenced by: bnj1128 32683 |
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