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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1083 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 32277. 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 |
---|---|
bnj1083.3 | ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) |
bnj1083.8 | ⊢ 𝐾 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} |
Ref | Expression |
---|---|
bnj1083 | ⊢ (𝑓 ∈ 𝐾 ↔ ∃𝑛𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rex 3144 | . 2 ⊢ (∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ∃𝑛(𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))) | |
2 | bnj1083.8 | . . 3 ⊢ 𝐾 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} | |
3 | 2 | abeq2i 2948 | . 2 ⊢ (𝑓 ∈ 𝐾 ↔ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) |
4 | bnj1083.3 | . . . 4 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) | |
5 | bnj252 31968 | . . . 4 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))) | |
6 | 4, 5 | bitri 277 | . . 3 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))) |
7 | 6 | exbii 1844 | . 2 ⊢ (∃𝑛𝜒 ↔ ∃𝑛(𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))) |
8 | 1, 3, 7 | 3bitr4i 305 | 1 ⊢ (𝑓 ∈ 𝐾 ↔ ∃𝑛𝜒) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∃wex 1776 ∈ wcel 2110 {cab 2799 ∃wrex 3139 Fn wfn 6344 ∧ w-bnj17 31951 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-3an 1085 df-tru 1536 df-ex 1777 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-rex 3144 df-bnj17 31952 |
This theorem is referenced by: bnj1121 32252 bnj1145 32260 |
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