Theorem r19.29vva 3062

Description: A commonly used pattern based on r19.29 3054, version with two restricted quantifiers. (Contributed by Thierry Arnoux, 26-Nov-2017.)

Hypotheses

Ref	Expression
r19.29vva.1	⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ 𝜓) → 𝜒)
r19.29vva.2	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓)

Assertion

Ref	Expression
r19.29vva	⊢ (𝜑 → 𝜒)

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

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

Proof of Theorem r19.29vva

Step	Hyp	Ref	Expression
1		r19.29vva.1	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ 𝜓) → 𝜒)
2	1	ex 449	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝜓 → 𝜒))
3	2	ralrimiva 2949	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ 𝐵 (𝜓 → 𝜒))
4	3	ralrimiva 2949	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝜒))
5		r19.29vva.2	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓)
6	4, 5	r19.29d2r 3061	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ((𝜓 → 𝜒) ∧ 𝜓))
7		pm3.35 609	. . . . 5 ⊢ ((𝜓 ∧ (𝜓 → 𝜒)) → 𝜒)
8	7	ancoms 468	. . . 4 ⊢ (((𝜓 → 𝜒) ∧ 𝜓) → 𝜒)
9	8	rexlimivw 3011	. . 3 ⊢ (∃𝑦 ∈ 𝐵 ((𝜓 → 𝜒) ∧ 𝜓) → 𝜒)
10	9	rexlimivw 3011	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ((𝜓 → 𝜒) ∧ 𝜓) → 𝜒)
11	6, 10	syl 17	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-ral 2901  df-rex 2902

This theorem is referenced by:  trust  21843  utoptop  21848  metustto  22168  restmetu  22185  tgbtwndiff  25201  legov  25280  legso  25294  tglnne  25323  tglndim0  25324  tglinethru  25331  tglnne0  25335  tglnpt2  25336  footex  25413  midex  25429  opptgdim2  25437  cgrane1  25504  cgrane2  25505  cgrane3  25506  cgrane4  25507  cgrahl1  25508  cgrahl2  25509  cgracgr  25510  cgratr  25515  cgrabtwn  25517  cgrahl  25518  dfcgra2  25521  sacgr  25522  acopyeu  25525  f1otrge  25552  archiabllem2c  29080  txomap  29229  qtophaus  29231  pstmfval  29267  eulerpartlemgvv  29765  irrapxlem4  36407