Theorem reximddv2 3215

Description: Double deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Thierry Arnoux, 15-Dec-2019.)

Hypotheses

Ref	Expression
reximddv2.1	⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ 𝜓) → 𝜒)
reximddv2.2	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓)

Assertion

Ref	Expression
reximddv2	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒)

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

Allowed substitution hints:   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥,𝑦)

Proof of Theorem reximddv2

Step	Hyp	Ref	Expression
1		reximddv2.1	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ 𝜓) → 𝜒)
2	1	ex 412	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝜓 → 𝜒))
3	2	reximdva 3168	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑦 ∈ 𝐵 𝜓 → ∃𝑦 ∈ 𝐵 𝜒))
4	3	impr 454	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝜓)) → ∃𝑦 ∈ 𝐵 𝜒)
5		reximddv2.2	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓)
6	4, 5	reximddv 3171	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108  ∃wrex 3070

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 3071

This theorem is referenced by:  r19.29vva  3216  r19.29vvaOLD  3217  prmgaplem8  17096  cpmadugsumfi  22883  cpmidg2sum  22886  cayhamlem4  22894  ltgseg  28604  cgraswap  28828  cgracom  28830  cgratr  28831  flatcgra  28832  dfcgra2  28838  xrofsup  32771  elrlocbasi  33270  aks6d1c2  42131  prmunb2  44330