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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-vexwt | Structured version Visualization version GIF version |
Description: Closed form of bj-vexw 32855. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.) Use bj-vexwvt 32856 instead when sufficient. (New usage is discouraged.) |
Ref | Expression |
---|---|
bj-vexwt | ⊢ (∀𝑥𝜑 → 𝑦 ∈ {𝑥 ∣ 𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | stdpc4 2353 | . 2 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑) | |
2 | df-clab 2609 | . 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑) | |
3 | 1, 2 | sylibr 224 | 1 ⊢ (∀𝑥𝜑 → 𝑦 ∈ {𝑥 ∣ 𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1481 [wsb 1880 ∈ wcel 1990 {cab 2608 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-12 2047 ax-13 2246 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 df-sb 1881 df-clab 2609 |
This theorem is referenced by: bj-vexw 32855 |
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