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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-abex | Structured version Visualization version GIF version |
Description: Two ways of stating that the extension of a formula is a set. (Contributed by BJ, 18-Jan-2025.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-abex | ⊢ ({𝑥 ∣ 𝜑} ∈ V ↔ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isset 3486 | . 2 ⊢ ({𝑥 ∣ 𝜑} ∈ V ↔ ∃𝑦 𝑦 = {𝑥 ∣ 𝜑}) | |
2 | eqabb 2872 | . . 3 ⊢ (𝑦 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑)) | |
3 | 2 | exbii 1849 | . 2 ⊢ (∃𝑦 𝑦 = {𝑥 ∣ 𝜑} ↔ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑)) |
4 | 1, 3 | bitri 275 | 1 ⊢ ({𝑥 ∣ 𝜑} ∈ V ↔ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∀wal 1538 = wceq 1540 ∃wex 1780 ∈ wcel 2105 {cab 2708 Vcvv 3473 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1543 df-ex 1781 df-nf 1785 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-v 3475 |
This theorem is referenced by: (None) |
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