Theorem reximddv2 3227

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 418	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝜓 → 𝜒))
3	2	reximdva 3181	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑦 ∈ 𝐵 𝜓 → ∃𝑦 ∈ 𝐵 𝜒))
4	3	impr 460	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝜓)) → ∃𝑦 ∈ 𝐵 𝜒)
5		reximddv2.2	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓)
6	4, 5	reximddv 3184	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2146  ∃wrex 3092

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-rex 3093

This theorem is used by:  r19.29vva  3228  prmgaplem8  17143  cpmadugsumfi  23071  cpmidg2sum  23074  cayhamlem4  23082  ltgseg  28902  cgraswap  29168  cgracom  29170  cgratr  29171  flatcgra  29172  dfcgra2  29178  xrofsup  33149  elrlocbasi  33618  aks6d1c2  42938  prmunb2  45062