Theorem bj-rexvwv 32060

Description: A weak version of rexv 3193 not using ax-ext 2590 (nor df-cleq 2603, df-clel 2606, df-v 3175) with an additional dv condition to avoid dependency on ax-13 2234 as well. See bj-ralvw 32059. (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 2902	. 2 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜓}𝜑 ↔ ∃𝑥(𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑))
2		bj-rexvwv.1	. . . . 5 ⊢ 𝜓
3	2	bj-vexwv 32051	. . . 4 ⊢ 𝑥 ∈ {𝑦 ∣ 𝜓}
4	3	biantrur 526	. . 3 ⊢ (𝜑 ↔ (𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑))
5	4	exbii 1764	. 2 ⊢ (∃𝑥𝜑 ↔ ∃𝑥(𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑))
6	1, 5	bitr4i 266	1 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜓}𝜑 ↔ ∃𝑥𝜑)

Syntax hints:   ↔ wb 195   ∧ wa 383  ∃wex 1695   ∈ wcel 1977  {cab 2596  ∃wrex 2897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-sb 1868  df-clab 2597  df-rex 2902

This theorem is referenced by: (None)