Theorem r19.41vv 3207

Description: Version of r19.41v 3167 with two quantifiers. (Contributed by Thierry Arnoux, 25-Jan-2017.)

Assertion

Ref	Expression
r19.41vv	⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ 𝜓))

Distinct variable groups:   𝜓,𝑥   𝜓,𝑦

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

Proof of Theorem r19.41vv

Step	Hyp	Ref	Expression
1		r19.41v 3167	. . 3 ⊢ (∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∃𝑦 ∈ 𝐵 𝜑 ∧ 𝜓))
2	1	rexbii 3084	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ ∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝜑 ∧ 𝜓))
3		r19.41v 3167	. 2 ⊢ (∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ 𝜓))
4	2, 3	bitri 275	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ 𝜓))

Syntax hints:   ↔ wb 206   ∧ wa 395  ∃wrex 3061

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-rex 3062

This theorem is referenced by:  genpass  10932  mulsuniflem  28141  addsdilem2  28144  mulsasslem1  28155  mulsasslem2  28156  dfcgra2  28898  axeuclid  29032  wspthsnwspthsnon  29984  dya2iocnrect  34425  satfv0  35540  satfv1  35545  satf0  35554  itg2addnclem3  37994  prprelprb  47977  prprspr2  47978