Theorem reximddv2 3192

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 3146	. . 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 3149	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2113  ∃wrex 3057

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-rex 3058

This theorem is referenced by:  r19.29vva  3193  prmgaplem8  16972  cpmadugsumfi  22793  cpmidg2sum  22796  cayhamlem4  22804  ltgseg  28575  cgraswap  28799  cgracom  28801  cgratr  28802  flatcgra  28803  dfcgra2  28809  xrofsup  32754  elrlocbasi  33240  aks6d1c2  42243  prmunb2  44428