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 3076	. 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 3053

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

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-rex 3054

This theorem is referenced by:  genpass  10962  mulsuniflem  28052  addsdilem2  28055  mulsasslem1  28066  mulsasslem2  28067  dfcgra2  28757  axeuclid  28890  wspthsnwspthsnon  29846  dya2iocnrect  34272  satfv0  35345  satfv1  35350  satf0  35359  itg2addnclem3  37667  prprelprb  47518  prprspr2  47519