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Theorem r19.29imd 3222
 Description: Theorem 19.29 of [Margaris] p. 90 with an implication in the hypothesis containing the generalization, deduction version. (Contributed by AV, 19-Jan-2019.)
Hypotheses
Ref Expression
r19.29imd.1 (𝜑 → ∃𝑥𝐴 𝜓)
r19.29imd.2 (𝜑 → ∀𝑥𝐴 (𝜓𝜒))
Assertion
Ref Expression
r19.29imd (𝜑 → ∃𝑥𝐴 (𝜓𝜒))

Proof of Theorem r19.29imd
StepHypRef Expression
1 r19.29imd.1 . . 3 (𝜑 → ∃𝑥𝐴 𝜓)
2 r19.29imd.2 . . 3 (𝜑 → ∀𝑥𝐴 (𝜓𝜒))
3 r19.29r 3221 . . 3 ((∃𝑥𝐴 𝜓 ∧ ∀𝑥𝐴 (𝜓𝜒)) → ∃𝑥𝐴 (𝜓 ∧ (𝜓𝜒)))
41, 2, 3syl2anc 573 . 2 (𝜑 → ∃𝑥𝐴 (𝜓 ∧ (𝜓𝜒)))
5 abai 823 . . 3 ((𝜓𝜒) ↔ (𝜓 ∧ (𝜓𝜒)))
65rexbii 3189 . 2 (∃𝑥𝐴 (𝜓𝜒) ↔ ∃𝑥𝐴 (𝜓 ∧ (𝜓𝜒)))
74, 6sylibr 224 1 (𝜑 → ∃𝑥𝐴 (𝜓𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382  ∀wral 3061  ∃wrex 3062 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885 This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853  df-ral 3066  df-rex 3067 This theorem is referenced by:  psgndif  20164  neik0pk1imk0  38871
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