| Mathbox for Jonathan Ben-Naim |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj591 | Structured version Visualization version GIF version | ||
| Description: Technical lemma for bnj852 34935. 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 |
|---|---|
| bnj591.1 | ⊢ (𝜃 ↔ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑗) = (𝑔‘𝑗))) |
| Ref | Expression |
|---|---|
| bnj591 | ⊢ ([𝑘 / 𝑗]𝜃 ↔ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑘) = (𝑔‘𝑘))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bnj591.1 | . . 3 ⊢ (𝜃 ↔ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑗) = (𝑔‘𝑗))) | |
| 2 | 1 | sbcbii 3846 | . 2 ⊢ ([𝑘 / 𝑗]𝜃 ↔ [𝑘 / 𝑗]((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑗) = (𝑔‘𝑗))) |
| 3 | vex 3484 | . . 3 ⊢ 𝑘 ∈ V | |
| 4 | fveq2 6906 | . . . . 5 ⊢ (𝑗 = 𝑘 → (𝑓‘𝑗) = (𝑓‘𝑘)) | |
| 5 | fveq2 6906 | . . . . 5 ⊢ (𝑗 = 𝑘 → (𝑔‘𝑗) = (𝑔‘𝑘)) | |
| 6 | 4, 5 | eqeq12d 2753 | . . . 4 ⊢ (𝑗 = 𝑘 → ((𝑓‘𝑗) = (𝑔‘𝑗) ↔ (𝑓‘𝑘) = (𝑔‘𝑘))) |
| 7 | 6 | imbi2d 340 | . . 3 ⊢ (𝑗 = 𝑘 → (((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑗) = (𝑔‘𝑗)) ↔ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑘) = (𝑔‘𝑘)))) |
| 8 | 3, 7 | sbcie 3830 | . 2 ⊢ ([𝑘 / 𝑗]((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑗) = (𝑔‘𝑗)) ↔ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑘) = (𝑔‘𝑘))) |
| 9 | 2, 8 | bitri 275 | 1 ⊢ ([𝑘 / 𝑗]𝜃 ↔ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑘) = (𝑔‘𝑘))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 [wsbc 3788 ‘cfv 6561 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 |
| This theorem is referenced by: bnj580 34927 |
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