Theorem bj-rexvw 37884

Description: A weak version of rexv 3477 not using ax-ext 2732 (nor df-cleq 2752, df-clel 2835, df-v 3452), and only core FOL axioms. See also bj-ralvw 37883. (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 3087	. 2 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜓}𝜑 ↔ ∃𝑥(𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑))
2		bj-rexvw.1	. . . . 5 ⊢ 𝜓
3	2	vexw 2744	. . . 4 ⊢ 𝑥 ∈ {𝑦 ∣ 𝜓}
4	3	biantrur 540	. . 3 ⊢ (𝜑 ↔ (𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑))
5	4	exbii 1880	. 2 ⊢ (∃𝑥𝜑 ↔ ∃𝑥(𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑))
6	1, 5	bitr4i 281	1 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜓}𝜑 ↔ ∃𝑥𝜑)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401  ∃wex 1811   ∈ wcel 2144  {cab 2738  ∃wrex 3086

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-sb 2099  df-clab 2739  df-rex 3087

This theorem is used by: (None)