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Theorem r19.29ffa 29629
Description: A commonly used pattern based on r19.29 3210, version with two restricted quantifiers. (Contributed by Thierry Arnoux, 26-Nov-2017.)
Hypothesis
Ref Expression
r19.29ffa.3 ((((𝜑𝑥𝐴) ∧ 𝑦𝐵) ∧ 𝜓) → 𝜒)
Assertion
Ref Expression
r19.29ffa ((𝜑 ∧ ∃𝑥𝐴𝑦𝐵 𝜓) → 𝜒)
Distinct variable groups:   𝑦,𝐴   𝜑,𝑥,𝑦   𝜒,𝑥,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥,𝑦)

Proof of Theorem r19.29ffa
StepHypRef Expression
1 r19.29ffa.3 . . . . . . 7 ((((𝜑𝑥𝐴) ∧ 𝑦𝐵) ∧ 𝜓) → 𝜒)
21ex 449 . . . . . 6 (((𝜑𝑥𝐴) ∧ 𝑦𝐵) → (𝜓𝜒))
32ralrimiva 3104 . . . . 5 ((𝜑𝑥𝐴) → ∀𝑦𝐵 (𝜓𝜒))
43ralrimiva 3104 . . . 4 (𝜑 → ∀𝑥𝐴𝑦𝐵 (𝜓𝜒))
54adantr 472 . . 3 ((𝜑 ∧ ∃𝑥𝐴𝑦𝐵 𝜓) → ∀𝑥𝐴𝑦𝐵 (𝜓𝜒))
6 simpr 479 . . 3 ((𝜑 ∧ ∃𝑥𝐴𝑦𝐵 𝜓) → ∃𝑥𝐴𝑦𝐵 𝜓)
75, 6r19.29d2r 3218 . 2 ((𝜑 ∧ ∃𝑥𝐴𝑦𝐵 𝜓) → ∃𝑥𝐴𝑦𝐵 ((𝜓𝜒) ∧ 𝜓))
8 pm3.35 612 . . . . 5 ((𝜓 ∧ (𝜓𝜒)) → 𝜒)
98ancoms 468 . . . 4 (((𝜓𝜒) ∧ 𝜓) → 𝜒)
109rexlimivw 3167 . . 3 (∃𝑦𝐵 ((𝜓𝜒) ∧ 𝜓) → 𝜒)
1110rexlimivw 3167 . 2 (∃𝑥𝐴𝑦𝐵 ((𝜓𝜒) ∧ 𝜓) → 𝜒)
127, 11syl 17 1 ((𝜑 ∧ ∃𝑥𝐴𝑦𝐵 𝜓) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2139  wral 3050  wrex 3051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854  df-ral 3055  df-rex 3056
This theorem is referenced by:  reprsuc  31002
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