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Theorem r19.29d2r 2472
Description: Theorem 19.29 of [Margaris] p. 90 with two restricted quantifiers, deduction version (Contributed by Thierry Arnoux, 30-Jan-2017.)
Hypotheses
Ref Expression
r19.29d2r.1 (𝜑 → ∀𝑥𝐴𝑦𝐵 𝜓)
r19.29d2r.2 (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜒)
Assertion
Ref Expression
r19.29d2r (𝜑 → ∃𝑥𝐴𝑦𝐵 (𝜓𝜒))

Proof of Theorem r19.29d2r
StepHypRef Expression
1 r19.29d2r.1 . . 3 (𝜑 → ∀𝑥𝐴𝑦𝐵 𝜓)
2 r19.29d2r.2 . . 3 (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜒)
3 r19.29 2467 . . 3 ((∀𝑥𝐴𝑦𝐵 𝜓 ∧ ∃𝑥𝐴𝑦𝐵 𝜒) → ∃𝑥𝐴 (∀𝑦𝐵 𝜓 ∧ ∃𝑦𝐵 𝜒))
41, 2, 3syl2anc 397 . 2 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 𝜓 ∧ ∃𝑦𝐵 𝜒))
5 r19.29 2467 . . 3 ((∀𝑦𝐵 𝜓 ∧ ∃𝑦𝐵 𝜒) → ∃𝑦𝐵 (𝜓𝜒))
65reximi 2433 . 2 (∃𝑥𝐴 (∀𝑦𝐵 𝜓 ∧ ∃𝑦𝐵 𝜒) → ∃𝑥𝐴𝑦𝐵 (𝜓𝜒))
74, 6syl 14 1 (𝜑 → ∃𝑥𝐴𝑦𝐵 (𝜓𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wral 2323  wrex 2324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-ral 2328  df-rex 2329
This theorem is referenced by:  r19.29vva  2473  cauappcvgprlemdisj  6807
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