Theorem bj-rexvwv 32866

Description: A weak version of rexv 3220 not using ax-ext 2602 (nor df-cleq 2615, df-clel 2618, df-v 3202) with an additional dv condition to avoid dependency on ax-13 2246 as well. See bj-ralvw 32865. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-rexvwv.1	⊢ 𝜓

Assertion

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

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem bj-rexvwv

Step	Hyp	Ref	Expression
1		df-rex 2918	. 2 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜓}𝜑 ↔ ∃𝑥(𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑))
2		bj-rexvwv.1	. . . . 5 ⊢ 𝜓
3	2	bj-vexwv 32857	. . . 4 ⊢ 𝑥 ∈ {𝑦 ∣ 𝜓}
4	3	biantrur 527	. . 3 ⊢ (𝜑 ↔ (𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑))
5	4	exbii 1774	. 2 ⊢ (∃𝑥𝜑 ↔ ∃𝑥(𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑))
6	1, 5	bitr4i 267	1 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜓}𝜑 ↔ ∃𝑥𝜑)

Syntax hints:   ↔ wb 196   ∧ wa 384  ∃wex 1704   ∈ wcel 1990  {cab 2608  ∃wrex 2913

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  df-rex 2918

This theorem is referenced by: (None)