Theorem ee4anvOLD 2386

Description: Obsolete version of ee4anv 2385 as of 26-Oct-2025. (Contributed by NM, 31-Jul-1995.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ee4anvOLD	⊢ (∃𝑥∃𝑦∃𝑧∃𝑤(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤𝜓))

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

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

Proof of Theorem ee4anvOLD

Step	Hyp	Ref	Expression
1		excom 2199	. . 3 ⊢ (∃𝑦∃𝑧∃𝑤(𝜑 ∧ 𝜓) ↔ ∃𝑧∃𝑦∃𝑤(𝜑 ∧ 𝜓))
2	1	exbii 1871	. 2 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤(𝜑 ∧ 𝜓) ↔ ∃𝑥∃𝑧∃𝑦∃𝑤(𝜑 ∧ 𝜓))
3		eeanv 2383	. . 3 ⊢ (∃𝑦∃𝑤(𝜑 ∧ 𝜓) ↔ (∃𝑦𝜑 ∧ ∃𝑤𝜓))
4	3	2exbii 1872	. 2 ⊢ (∃𝑥∃𝑧∃𝑦∃𝑤(𝜑 ∧ 𝜓) ↔ ∃𝑥∃𝑧(∃𝑦𝜑 ∧ ∃𝑤𝜓))
5		eeanv 2383	. 2 ⊢ (∃𝑥∃𝑧(∃𝑦𝜑 ∧ ∃𝑤𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤𝜓))
6	2, 4, 5	3bitri 300	1 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤𝜓))

Syntax hints:   ↔ wb 209   ∧ wa 400  ∃wex 1802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-10 2178  ax-11 2194  ax-12 2215

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1803  df-nf 1807

This theorem is referenced by: (None)