Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > vexwt | Structured version Visualization version GIF version |
Description: A standard theorem of predicate calculus (stdpc4 2073) expressed using class abstractions. Closed form of vexw 2805. (Contributed by BJ, 14-Jun-2019.) |
Ref | Expression |
---|---|
vexwt | ⊢ (∀𝑥𝜑 → 𝑦 ∈ {𝑥 ∣ 𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | stdpc4 2073 | . 2 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑) | |
2 | df-clab 2800 | . 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑) | |
3 | 1, 2 | sylibr 236 | 1 ⊢ (∀𝑥𝜑 → 𝑦 ∈ {𝑥 ∣ 𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1535 [wsb 2069 ∈ wcel 2114 {cab 2799 |
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 1911 |
This theorem depends on definitions: df-bi 209 df-sb 2070 df-clab 2800 |
This theorem is referenced by: bj-issetwt 34205 bj-abv 34239 |
Copyright terms: Public domain | W3C validator |