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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 3851 | . 2 ⊢ ([𝑍 / 𝑦]𝜑 ↔ [𝑍 / 𝑦](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) |
3 | bnj91.2 | . . 3 ⊢ 𝑍 ∈ V | |
4 | 3 | bnj525 34730 | . 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 1536 ∈ wcel 2105 Vcvv 3477 [wsbc 3790 ∅c0 4338 ‘cfv 6562 predc-bnj14 34680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-sbc 3791 |
This theorem is referenced by: bnj118 34861 bnj125 34864 |
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