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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 31996 | . . 3 ⊢ 1o ∈ V | |
4 | 2, 3 | bnj91 32135 | . 2 ⊢ ([1o / 𝑛]𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) |
5 | 1, 4 | bitri 277 | 1 ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 [wsbc 3774 ∅c0 4293 ‘cfv 6357 1oc1o 8097 predc-bnj14 31960 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-pw 4543 df-sn 4570 df-suc 6199 df-1o 8104 |
This theorem is referenced by: bnj151 32151 bnj153 32154 |
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