Theorem bj-rexvw 36822

Description: A weak version of rexv 3493 not using ax-ext 2706 (nor df-cleq 2726, df-clel 2808, df-v 3466), and only core FOL axioms. See also bj-ralvw 36821. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-rexvw.1	⊢ 𝜓

Assertion

Ref	Expression
bj-rexvw	⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜓}𝜑 ↔ ∃𝑥𝜑)

Proof of Theorem bj-rexvw

Step	Hyp	Ref	Expression
1		df-rex 3060	. 2 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜓}𝜑 ↔ ∃𝑥(𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑))
2		bj-rexvw.1	. . . . 5 ⊢ 𝜓
3	2	vexw 2718	. . . 4 ⊢ 𝑥 ∈ {𝑦 ∣ 𝜓}
4	3	biantrur 530	. . 3 ⊢ (𝜑 ↔ (𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑))
5	4	exbii 1847	. 2 ⊢ (∃𝑥𝜑 ↔ ∃𝑥(𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑))
6	1, 5	bitr4i 278	1 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜓}𝜑 ↔ ∃𝑥𝜑)

Syntax hints:   ↔ wb 206   ∧ wa 395  ∃wex 1778   ∈ wcel 2107  {cab 2712  ∃wrex 3059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-sb 2064  df-clab 2713  df-rex 3060

This theorem is referenced by: (None)