Theorem r19.44zv 4469

Description: Restricted version of Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)

Assertion

Ref	Expression
r19.44zv	⊢ (𝐴 ≠ ∅ → (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ 𝜓)))

Distinct variable groups:   𝑥,𝐴   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem r19.44zv

Step	Hyp	Ref	Expression
1		r19.43 3132	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))
2		r19.9rzv 4465	. . 3 ⊢ (𝐴 ≠ ∅ → (𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜓))
3	2	orbi2d 928	. 2 ⊢ (𝐴 ≠ ∅ → ((∃𝑥 ∈ 𝐴 𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓)))
4	1, 3	bitr4id 293	1 ⊢ (𝐴 ≠ ∅ → (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ 𝜓)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∨ wo 860   ≠ wne 2957  ∃wrex 3088  ∅c0 4285

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-ne 2958  df-ral 3079  df-rex 3089  df-dif 3907  df-nul 4286

This theorem is used by:  fmla1  35887