Theorem bj-vexwv 32857

Description: Version of bj-vexw 32855 with a dv condition, which does not require ax-13 2246. The degenerate instance of bj-vexw 32855 is a simple consequence of abid 2610 (which does not depend on ax-13 2246 either). (Contributed by BJ, 13-Jun-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-vexwv.1	⊢ 𝜑

Assertion

Ref	Expression
bj-vexwv	⊢ 𝑦 ∈ {𝑥 ∣ 𝜑}

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bj-vexwv

Step	Hyp	Ref	Expression
1		bj-vexwvt 32856	. 2 ⊢ (∀𝑥𝜑 → 𝑦 ∈ {𝑥 ∣ 𝜑})
2		bj-vexwv.1	. 2 ⊢ 𝜑
3	1, 2	mpg 1724	1 ⊢ 𝑦 ∈ {𝑥 ∣ 𝜑}

Syntax hints:   ∈ 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-denotes  32858  bj-rexvwv  32866  bj-rababwv  32867  bj-df-v  33016