| Mathbox for Jonathan Ben-Naim |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj934 | Structured version Visualization version GIF version | ||
| Description: Technical lemma for bnj69 35024. 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 |
|---|---|
| bnj934.1 | ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) |
| bnj934.4 | ⊢ (𝜑′ ↔ [𝑝 / 𝑛]𝜑) |
| bnj934.7 | ⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑′) |
| bnj934.50 | ⊢ 𝐺 ∈ V |
| Ref | Expression |
|---|---|
| bnj934 | ⊢ ((𝜑 ∧ (𝐺‘∅) = (𝑓‘∅)) → 𝜑″) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bnj934.1 | . . . 4 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) | |
| 2 | eqtr 2760 | . . . 4 ⊢ (((𝐺‘∅) = (𝑓‘∅) ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) → (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅)) | |
| 3 | 1, 2 | sylan2b 594 | . . 3 ⊢ (((𝐺‘∅) = (𝑓‘∅) ∧ 𝜑) → (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅)) |
| 4 | bnj934.7 | . . . . 5 ⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑′) | |
| 5 | bnj934.4 | . . . . . . . 8 ⊢ (𝜑′ ↔ [𝑝 / 𝑛]𝜑) | |
| 6 | vex 3484 | . . . . . . . 8 ⊢ 𝑝 ∈ V | |
| 7 | 1, 5, 6 | bnj523 34901 | . . . . . . 7 ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) |
| 8 | 7, 1 | bitr4i 278 | . . . . . 6 ⊢ (𝜑′ ↔ 𝜑) |
| 9 | 8 | sbcbii 3846 | . . . . 5 ⊢ ([𝐺 / 𝑓]𝜑′ ↔ [𝐺 / 𝑓]𝜑) |
| 10 | 4, 9 | bitri 275 | . . . 4 ⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑) |
| 11 | bnj934.50 | . . . 4 ⊢ 𝐺 ∈ V | |
| 12 | 1, 10, 11 | bnj609 34931 | . . 3 ⊢ (𝜑″ ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅)) |
| 13 | 3, 12 | sylibr 234 | . 2 ⊢ (((𝐺‘∅) = (𝑓‘∅) ∧ 𝜑) → 𝜑″) |
| 14 | 13 | ancoms 458 | 1 ⊢ ((𝜑 ∧ (𝐺‘∅) = (𝑓‘∅)) → 𝜑″) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = 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: bnj929 34950 |
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