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Theorem r19.44mv 3357
Description: Restricted version of Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
Assertion
Ref Expression
r19.44mv (∃𝑦 𝑦𝐴 → (∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓)))
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)

Proof of Theorem r19.44mv
StepHypRef Expression
1 r19.9rmv 3354 . . 3 (∃𝑦 𝑦𝐴 → (𝜓 ↔ ∃𝑥𝐴 𝜓))
21orbi2d 737 . 2 (∃𝑦 𝑦𝐴 → ((∃𝑥𝐴 𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓)))
3 r19.43 2518 . 2 (∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓))
42, 3syl6rbbr 197 1 (∃𝑦 𝑦𝐴 → (∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wo 662  wex 1422  wcel 1434  wrex 2354
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-cleq 2076  df-clel 2079  df-rex 2359
This theorem is referenced by:  frecabcl  6094
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