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Theorem r19.45mv 3343
Description: Restricted version of Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
Assertion
Ref Expression
r19.45mv (∃𝑥 𝑥𝐴 → (∃𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∨ ∃𝑥𝐴 𝜓)))
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem r19.45mv
StepHypRef Expression
1 r19.9rmv 3341 . . 3 (∃𝑥 𝑥𝐴 → (𝜑 ↔ ∃𝑥𝐴 𝜑))
21orbi1d 715 . 2 (∃𝑥 𝑥𝐴 → ((𝜑 ∨ ∃𝑥𝐴 𝜓) ↔ (∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓)))
3 r19.43 2485 . 2 (∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓))
42, 3syl6rbbr 192 1 (∃𝑥 𝑥𝐴 → (∃𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∨ ∃𝑥𝐴 𝜓)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wo 639  wex 1397  wcel 1409  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-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-cleq 2049  df-clel 2052  df-rex 2329
This theorem is referenced by:  ltexprlemloc  6763
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