Theorem r19.41vv 3222

Description: Version of r19.41v 3186 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 3186	. . 3 ⊢ (∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∃𝑦 ∈ 𝐵 𝜑 ∧ 𝜓))
2	1	rexbii 3091	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ ∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝜑 ∧ 𝜓))
3		r19.41v 3186	. 2 ⊢ (∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ 𝜓))
4	2, 3	bitri 274	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ 𝜓))

Syntax hints:   ↔ wb 205   ∧ wa 394  ∃wrex 3067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905

This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1774  df-rex 3068

This theorem is referenced by:  genpass  11040  mulsuniflem  28069  addsdilem2  28072  mulsasslem1  28083  mulsasslem2  28084  dfcgra2  28654  axeuclid  28794  wspthsnwspthsnon  29747  dya2iocnrect  33934  satfv0  35001  satfv1  35006  satf0  35015  itg2addnclem3  37179  prprelprb  46886  prprspr2  46887