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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1454 | 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 |
---|---|
bnj1454.1 | ⊢ 𝐴 = {𝑥 ∣ 𝜑} |
Ref | Expression |
---|---|
bnj1454 | ⊢ (𝐵 ∈ V → (𝐵 ∈ 𝐴 ↔ [𝐵 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sbc 3634 | . . 3 ⊢ ([𝐵 / 𝑥]𝜑 ↔ 𝐵 ∈ {𝑥 ∣ 𝜑}) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐵 ∈ V → ([𝐵 / 𝑥]𝜑 ↔ 𝐵 ∈ {𝑥 ∣ 𝜑})) |
3 | bnj1454.1 | . . 3 ⊢ 𝐴 = {𝑥 ∣ 𝜑} | |
4 | 3 | eleq2i 2870 | . 2 ⊢ (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ {𝑥 ∣ 𝜑}) |
5 | 2, 4 | syl6rbbr 282 | 1 ⊢ (𝐵 ∈ V → (𝐵 ∈ 𝐴 ↔ [𝐵 / 𝑥]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1653 ∈ wcel 2157 {cab 2785 Vcvv 3385 [wsbc 3633 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-ext 2777 |
This theorem depends on definitions: df-bi 199 df-an 386 df-ex 1876 df-cleq 2792 df-clel 2795 df-sbc 3634 |
This theorem is referenced by: bnj1452 31637 bnj1463 31640 |
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