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| Description: Technical lemma for bnj150 34890. 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 | 
|---|---|
| bnj125.1 | ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) | 
| bnj125.2 | ⊢ (𝜑′ ↔ [1o / 𝑛]𝜑) | 
| bnj125.3 | ⊢ (𝜑″ ↔ [𝐹 / 𝑓]𝜑′) | 
| bnj125.4 | ⊢ 𝐹 = {〈∅, pred(𝑥, 𝐴, 𝑅)〉} | 
| Ref | Expression | 
|---|---|
| bnj125 | ⊢ (𝜑″ ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | bnj125.3 | . 2 ⊢ (𝜑″ ↔ [𝐹 / 𝑓]𝜑′) | |
| 2 | bnj125.2 | . . . 4 ⊢ (𝜑′ ↔ [1o / 𝑛]𝜑) | |
| 3 | 2 | sbcbii 3846 | . . 3 ⊢ ([𝐹 / 𝑓]𝜑′ ↔ [𝐹 / 𝑓][1o / 𝑛]𝜑) | 
| 4 | bnj125.1 | . . . . . 6 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) | |
| 5 | bnj105 34738 | . . . . . 6 ⊢ 1o ∈ V | |
| 6 | 4, 5 | bnj91 34875 | . . . . 5 ⊢ ([1o / 𝑛]𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) | 
| 7 | 6 | sbcbii 3846 | . . . 4 ⊢ ([𝐹 / 𝑓][1o / 𝑛]𝜑 ↔ [𝐹 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) | 
| 8 | bnj125.4 | . . . . . 6 ⊢ 𝐹 = {〈∅, pred(𝑥, 𝐴, 𝑅)〉} | |
| 9 | 8 | bnj95 34878 | . . . . 5 ⊢ 𝐹 ∈ V | 
| 10 | fveq1 6905 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘∅) = (𝐹‘∅)) | |
| 11 | 10 | eqeq1d 2739 | . . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))) | 
| 12 | 9, 11 | sbcie 3830 | . . . 4 ⊢ ([𝐹 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅)) | 
| 13 | 7, 12 | bitri 275 | . . 3 ⊢ ([𝐹 / 𝑓][1o / 𝑛]𝜑 ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅)) | 
| 14 | 3, 13 | bitri 275 | . 2 ⊢ ([𝐹 / 𝑓]𝜑′ ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅)) | 
| 15 | 1, 14 | bitri 275 | 1 ⊢ (𝜑″ ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 = wceq 1540 [wsbc 3788 ∅c0 4333 {csn 4626 〈cop 4632 ‘cfv 6561 1oc1o 8499 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 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-pw 4602 df-sn 4627 df-pr 4629 df-uni 4908 df-br 5144 df-suc 6390 df-iota 6514 df-fv 6569 df-1o 8506 | 
| This theorem is referenced by: bnj150 34890 bnj153 34894 | 
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