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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj118 | 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 |
---|---|
bnj118.1 | ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) |
bnj118.2 | ⊢ (𝜑′ ↔ [1o / 𝑛]𝜑) |
Ref | Expression |
---|---|
bnj118 | ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj118.2 | . 2 ⊢ (𝜑′ ↔ [1o / 𝑛]𝜑) | |
2 | bnj118.1 | . . 3 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) | |
3 | bnj105 32689 | . . 3 ⊢ 1o ∈ V | |
4 | 2, 3 | bnj91 32827 | . 2 ⊢ ([1o / 𝑛]𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) |
5 | 1, 4 | bitri 274 | 1 ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1539 [wsbc 3716 ∅c0 4257 ‘cfv 6427 1oc1o 8278 predc-bnj14 32653 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 ax-sep 5222 ax-nul 5229 ax-pow 5287 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-v 3432 df-sbc 3717 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4258 df-pw 4536 df-sn 4563 df-suc 6266 df-1o 8285 |
This theorem is referenced by: bnj151 32843 bnj153 32846 |
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