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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-abvALT | Structured version Visualization version GIF version | ||
| Description: Alternate version of bj-abv 36885; shorter but uses ax-8 2110. (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| bj-abvALT | ⊢ (∀𝑥𝜑 → {𝑥 ∣ 𝜑} = V) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ax-5 1910 | . . 3 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑) | |
| 2 | vexwt 2718 | . . 3 ⊢ (∀𝑥𝜑 → 𝑦 ∈ {𝑥 ∣ 𝜑}) | |
| 3 | 1, 2 | alrimih 1824 | . 2 ⊢ (∀𝑥𝜑 → ∀𝑦 𝑦 ∈ {𝑥 ∣ 𝜑}) |
| 4 | eqv 3489 | . 2 ⊢ ({𝑥 ∣ 𝜑} = V ↔ ∀𝑦 𝑦 ∈ {𝑥 ∣ 𝜑}) | |
| 5 | 3, 4 | sylibr 234 | 1 ⊢ (∀𝑥𝜑 → {𝑥 ∣ 𝜑} = V) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∀wal 1538 = wceq 1540 ∈ wcel 2108 {cab 2713 Vcvv 3479 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2707 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 |
| This theorem is referenced by: (None) |
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