Theorem bj-vexwvt 32856

Description: Closed form of bj-vexwv 32857 and version of bj-vexwt 32854 with a dv condition, which does not require ax-13 2246. (Contributed by BJ, 13-Jun-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-vexwvt	⊢ (∀𝑥𝜑 → 𝑦 ∈ {𝑥 ∣ 𝜑})

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bj-vexwvt

Step	Hyp	Ref	Expression
1		bj-stdpc4v 32754	. 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})

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

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-vexwv  32857  bj-issetwt  32859  bj-abv  32901