Theorem rexcom4b 3482

Description: Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)

Hypothesis

Ref	Expression
rexcom4b.1	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
rexcom4b	⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 𝜑)

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

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

Proof of Theorem rexcom4b

Step	Hyp	Ref	Expression
1		rexcom4a 3269	. 2 ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ ∃𝑥 𝑥 = 𝐵))
2		rexcom4b.1	. . . . 5 ⊢ 𝐵 ∈ V
3	2	isseti 3468	. . . 4 ⊢ ∃𝑥 𝑥 = 𝐵
4	3	biantru 529	. . 3 ⊢ (𝜑 ↔ (𝜑 ∧ ∃𝑥 𝑥 = 𝐵))
5	4	rexbii 3077	. 2 ⊢ (∃𝑦 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ ∃𝑥 𝑥 = 𝐵))
6	1, 5	bitr4i 278	1 ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 𝜑)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∃wrex 3054  Vcvv 3450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-11 2158

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-clel 2804  df-rex 3055

This theorem is referenced by:  islshpat  39017