Theorem bj-rexcom4bv 37372

Description: Version of rexcom4b 3486 and bj-rexcom4b 37373 with a disjoint variable condition on 𝑥, 𝑉, hence removing dependency on df-sb 2092 and df-clab 2742 (so that it depends on df-clel 2838 and df-rex 3088 only on top of first-order logic). Prefer its use over bj-rexcom4b 37373 when sufficient (in particular when 𝑉 is substituted for V). Note the 𝑉 in the hypothesis instead of V. (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-rexcom4bv.1	⊢ 𝐵 ∈ 𝑉

Assertion

Ref	Expression
bj-rexcom4bv	⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 𝜑)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑉   𝑥,𝑦   𝜑,𝑥

Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)   𝐵(𝑦)   𝑉(𝑦)

Proof of Theorem bj-rexcom4bv

Step	Hyp	Ref	Expression
1		rexcom4a 3293	. 2 ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ ∃𝑥 𝑥 = 𝐵))
2		bj-rexcom4bv.1	. . . . 5 ⊢ 𝐵 ∈ 𝑉
3	2	bj-issetiv 37367	. . . 4 ⊢ ∃𝑥 𝑥 = 𝐵
4	3	biantru 537	. . 3 ⊢ (𝜑 ↔ (𝜑 ∧ ∃𝑥 𝑥 = 𝐵))
5	4	rexbii 3110	. 2 ⊢ (∃𝑦 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ ∃𝑥 𝑥 = 𝐵))
6	1, 5	bitr4i 280	1 ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 𝜑)

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1561  ∃wex 1800   ∈ wcel 2143  ∃wrex 3087

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-11 2192

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1801  df-clel 2838  df-rex 3088

This theorem is referenced by: (None)