Theorem bj-eeanvw 34895

Description: Version of exdistrv 1959 with a disjoint variable condition on 𝑥, 𝑦 not requiring ax-11 2154. (The same can be done with eeeanv 2348 and ee4anv 2349.) (Contributed by BJ, 29-Sep-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-eeanvw	⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))

Distinct variable groups:   𝜑,𝑦   𝜓,𝑥   𝑥,𝑦

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

Proof of Theorem bj-eeanvw

Step	Hyp	Ref	Expression
1		19.42v 1957	. . 3 ⊢ (∃𝑦(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑦𝜓))
2	1	exbii 1850	. 2 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓))
3		19.41v 1953	. 2 ⊢ (∃𝑥(𝜑 ∧ ∃𝑦𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
4	2, 3	bitri 274	1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))

Syntax hints:   ↔ wb 205   ∧ wa 396  ∃wex 1782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783

This theorem is referenced by: (None)