| Mathbox for Jonathan Ben-Naim |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj91 | Structured version Visualization version GIF version | ||
| Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| bnj91.1 | ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) |
| bnj91.2 | ⊢ 𝑍 ∈ V |
| Ref | Expression |
|---|---|
| bnj91 | ⊢ ([𝑍 / 𝑦]𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bnj91.1 | . . 3 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) | |
| 2 | 1 | sbcbii 3846 | . 2 ⊢ ([𝑍 / 𝑦]𝜑 ↔ [𝑍 / 𝑦](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) |
| 3 | bnj91.2 | . . 3 ⊢ 𝑍 ∈ V | |
| 4 | 3 | bnj525 34752 | . 2 ⊢ ([𝑍 / 𝑦](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) |
| 5 | 2, 4 | bitri 275 | 1 ⊢ ([𝑍 / 𝑦]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: bnj118 34883 bnj125 34886 |
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