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Theorem r19.29vvaOLD 3320
 Description: Obsolete version of r19.29vva 3319 as of 28-Jun-2023. (Contributed by Thierry Arnoux, 26-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
r19.29vva.1 ((((𝜑𝑥𝐴) ∧ 𝑦𝐵) ∧ 𝜓) → 𝜒)
r19.29vva.2 (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜓)
Assertion
Ref Expression
r19.29vvaOLD (𝜑𝜒)
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 r19.29vvaOLD
StepHypRef Expression
1 r19.29vva.1 . . . . . 6 ((((𝜑𝑥𝐴) ∧ 𝑦𝐵) ∧ 𝜓) → 𝜒)
21ex 416 . . . . 5 (((𝜑𝑥𝐴) ∧ 𝑦𝐵) → (𝜓𝜒))
32ralrimiva 3174 . . . 4 ((𝜑𝑥𝐴) → ∀𝑦𝐵 (𝜓𝜒))
43ralrimiva 3174 . . 3 (𝜑 → ∀𝑥𝐴𝑦𝐵 (𝜓𝜒))
5 r19.29vva.2 . . 3 (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜓)
64, 5r19.29d2r 3318 . 2 (𝜑 → ∃𝑥𝐴𝑦𝐵 ((𝜓𝜒) ∧ 𝜓))
7 pm3.35 802 . . . . 5 ((𝜓 ∧ (𝜓𝜒)) → 𝜒)
87ancoms 462 . . . 4 (((𝜓𝜒) ∧ 𝜓) → 𝜒)
98rexlimivw 3268 . . 3 (∃𝑦𝐵 ((𝜓𝜒) ∧ 𝜓) → 𝜒)
109rexlimivw 3268 . 2 (∃𝑥𝐴𝑦𝐵 ((𝜓𝜒) ∧ 𝜓) → 𝜒)
116, 10syl 17 1 (𝜑𝜒)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2114  ∀wral 3130  ∃wrex 3131 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-ral 3135  df-rex 3136 This theorem is referenced by: (None)
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