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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1538 | 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 |
---|---|
bnj1538.1 | ⊢ 𝐴 = {𝑥 ∈ 𝐵 ∣ 𝜑} |
Ref | Expression |
---|---|
bnj1538 | ⊢ (𝑥 ∈ 𝐴 → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1538.1 | . . 3 ⊢ 𝐴 = {𝑥 ∈ 𝐵 ∣ 𝜑} | |
2 | 1 | rabeq2i 3410 | . 2 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐵 ∧ 𝜑)) |
3 | 2 | simprbi 492 | 1 ⊢ (𝑥 ∈ 𝐴 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 {crab 3121 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-12 2220 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-tru 1660 df-ex 1879 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-rab 3126 |
This theorem is referenced by: bnj1279 31621 bnj1311 31627 bnj1418 31643 bnj1312 31661 bnj1523 31674 |
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