Description: Technical lemma for bnj69 . 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 | ||
|---|---|---|---|
| Hypotheses | bnj1083.3 | ⊢ ( 𝜒 ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) | |
| bnj1083.8 | ⊢ 𝐾 = { 𝑓 ∣ ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) } | ||
| Assertion | bnj1083 | ⊢ ( 𝑓 ∈ 𝐾 ↔ ∃ 𝑛 𝜒 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bnj1083.3 | ⊢ ( 𝜒 ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) | |
| 2 | bnj1083.8 | ⊢ 𝐾 = { 𝑓 ∣ ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) } | |
| 3 | df-rex | ⊢ ( ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ↔ ∃ 𝑛 ( 𝑛 ∈ 𝐷 ∧ ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) ) | |
| 4 | 2 | abeq2i | ⊢ ( 𝑓 ∈ 𝐾 ↔ ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) |
| 5 | bnj252 | ⊢ ( ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ↔ ( 𝑛 ∈ 𝐷 ∧ ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) ) | |
| 6 | 1 5 | bitri | ⊢ ( 𝜒 ↔ ( 𝑛 ∈ 𝐷 ∧ ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) ) |
| 7 | 6 | exbii | ⊢ ( ∃ 𝑛 𝜒 ↔ ∃ 𝑛 ( 𝑛 ∈ 𝐷 ∧ ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) ) |
| 8 | 3 4 7 | 3bitr4i | ⊢ ( 𝑓 ∈ 𝐾 ↔ ∃ 𝑛 𝜒 ) |