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| 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 | 
|---|---|
| bnj523.1 | ⊢ (𝜑 ↔ (𝐹‘∅) = pred(𝑋, 𝐴, 𝑅)) | 
| bnj523.2 | ⊢ (𝜑′ ↔ [𝑀 / 𝑛]𝜑) | 
| bnj523.3 | ⊢ 𝑀 ∈ V | 
| Ref | Expression | 
|---|---|
| bnj523 | ⊢ (𝜑′ ↔ (𝐹‘∅) = pred(𝑋, 𝐴, 𝑅)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | bnj523.2 | . 2 ⊢ (𝜑′ ↔ [𝑀 / 𝑛]𝜑) | |
| 2 | bnj523.1 | . . 3 ⊢ (𝜑 ↔ (𝐹‘∅) = pred(𝑋, 𝐴, 𝑅)) | |
| 3 | 2 | sbcbii 3846 | . 2 ⊢ ([𝑀 / 𝑛]𝜑 ↔ [𝑀 / 𝑛](𝐹‘∅) = pred(𝑋, 𝐴, 𝑅)) | 
| 4 | bnj523.3 | . . 3 ⊢ 𝑀 ∈ V | |
| 5 | 4 | bnj525 34752 | . 2 ⊢ ([𝑀 / 𝑛](𝐹‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝐹‘∅) = pred(𝑋, 𝐴, 𝑅)) | 
| 6 | 1, 3, 5 | 3bitri 297 | 1 ⊢ (𝜑′ ↔ (𝐹‘∅) = 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-sbc 3789 | 
| This theorem is referenced by: bnj600 34933 bnj908 34945 bnj934 34949 | 
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