Theorem bj-rexvw 37318

Description: A weak version of rexv 3480 not using ax-ext 2733 (nor df-cleq 2753, df-clel 2836, df-v 3455), and only core FOL axioms. See also bj-ralvw 37317. (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 3086	. 2 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜓}𝜑 ↔ ∃𝑥(𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑))
2		bj-rexvw.1	. . . . 5 ⊢ 𝜓
3	2	vexw 2745	. . . 4 ⊢ 𝑥 ∈ {𝑦 ∣ 𝜓}
4	3	biantrur 538	. . 3 ⊢ (𝜑 ↔ (𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑))
5	4	exbii 1867	. 2 ⊢ (∃𝑥𝜑 ↔ ∃𝑥(𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑))
6	1, 5	bitr4i 280	1 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜓}𝜑 ↔ ∃𝑥𝜑)

Syntax hints:   ↔ wb 208   ∧ wa 399  ∃wex 1798   ∈ wcel 2141  {cab 2739  ∃wrex 3085

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1799  df-sb 2090  df-clab 2740  df-rex 3086

This theorem is referenced by: (None)