Theorem r19.45mv 3508

Description: Restricted version of Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)

Assertion

Ref	Expression
r19.45mv	⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓)))

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem r19.45mv

Step	Hyp	Ref	Expression
1		r19.43 2628	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))
2		r19.9rmv 3506	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜑))
3	2	orbi1d 786	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ((𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓)))
4	1, 3	bitr4id 198	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓)))

Syntax hints:   → wi 4   ↔ wb 104   ∨ wo 703  ∃wex 1485   ∈ wcel 2141  ∃wrex 2449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-cleq 2163  df-clel 2166  df-rex 2454

This theorem is referenced by:  ltexprlemloc  7569