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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 36384; shorter but uses ax-8 2101. (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 1906 | . . 3 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑) | |
2 | vexwt 2710 | . . 3 ⊢ (∀𝑥𝜑 → 𝑦 ∈ {𝑥 ∣ 𝜑}) | |
3 | 1, 2 | alrimih 1819 | . 2 ⊢ (∀𝑥𝜑 → ∀𝑦 𝑦 ∈ {𝑥 ∣ 𝜑}) |
4 | eqv 3480 | . 2 ⊢ ({𝑥 ∣ 𝜑} = V ↔ ∀𝑦 𝑦 ∈ {𝑥 ∣ 𝜑}) | |
5 | 3, 4 | sylibr 233 | 1 ⊢ (∀𝑥𝜑 → {𝑥 ∣ 𝜑} = V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1532 = wceq 1534 ∈ wcel 2099 {cab 2705 Vcvv 3471 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3473 |
This theorem is referenced by: (None) |
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