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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1230 | 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 |
---|---|
bnj1230.1 | ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑} |
Ref | Expression |
---|---|
bnj1230 | ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1230.1 | . . 3 ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑} | |
2 | nfrab1 3384 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜑} | |
3 | 1, 2 | nfcxfr 2975 | . 2 ⊢ Ⅎ𝑥𝐵 |
4 | 3 | nfcrii 2970 | 1 ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1535 = wceq 1537 ∈ wcel 2114 {crab 3142 |
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 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 |
This theorem is referenced by: bnj1312 32330 |
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