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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj609 | 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 |
|---|---|
| bnj609.1 | ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) |
| bnj609.2 | ⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑) |
| bnj609.3 | ⊢ 𝐺 ∈ V |
| Ref | Expression |
|---|---|
| bnj609 | ⊢ (𝜑″ ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bnj609.2 | . 2 ⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑) | |
| 2 | bnj609.3 | . . 3 ⊢ 𝐺 ∈ V | |
| 3 | dfsbcq 3790 | . . 3 ⊢ (𝑒 = 𝐺 → ([𝑒 / 𝑓]𝜑 ↔ [𝐺 / 𝑓]𝜑)) | |
| 4 | fveq1 6905 | . . . 4 ⊢ (𝑒 = 𝐺 → (𝑒‘∅) = (𝐺‘∅)) | |
| 5 | 4 | eqeq1d 2739 | . . 3 ⊢ (𝑒 = 𝐺 → ((𝑒‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))) |
| 6 | bnj609.1 | . . . . 5 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) | |
| 7 | 6 | sbcbii 3846 | . . . 4 ⊢ ([𝑒 / 𝑓]𝜑 ↔ [𝑒 / 𝑓](𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) |
| 8 | vex 3484 | . . . . 5 ⊢ 𝑒 ∈ V | |
| 9 | fveq1 6905 | . . . . . 6 ⊢ (𝑓 = 𝑒 → (𝑓‘∅) = (𝑒‘∅)) | |
| 10 | 9 | eqeq1d 2739 | . . . . 5 ⊢ (𝑓 = 𝑒 → ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝑒‘∅) = pred(𝑋, 𝐴, 𝑅))) |
| 11 | 8, 10 | sbcie 3830 | . . . 4 ⊢ ([𝑒 / 𝑓](𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝑒‘∅) = pred(𝑋, 𝐴, 𝑅)) |
| 12 | 7, 11 | bitri 275 | . . 3 ⊢ ([𝑒 / 𝑓]𝜑 ↔ (𝑒‘∅) = pred(𝑋, 𝐴, 𝑅)) |
| 13 | 2, 3, 5, 12 | vtoclb 3568 | . 2 ⊢ ([𝐺 / 𝑓]𝜑 ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅)) |
| 14 | 1, 13 | bitri 275 | 1 ⊢ (𝜑″ ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 ∈ wcel 2108 Vcvv 3480 [wsbc 3788 ∅c0 4333 ‘cfv 6561 predc-bnj14 34702 |
| 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-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-sbc 3789 df-ss 3968 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 |
| This theorem is referenced by: bnj600 34933 bnj908 34945 bnj934 34949 |
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