Theorem r19.45av 2630

Description: Restricted version of one direction of Theorem 19.45 of [Margaris] p. 90. (The other direction doesn't hold when 𝐴 is empty.) (Contributed by NM, 2-Apr-2004.)

Assertion

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

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem r19.45av

Step	Hyp	Ref	Expression
1		r19.43 2628	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))
2		idd 21	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜑))
3	2	rexlimiv 2581	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜑)
4	3	orim1i 755	. 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓) → (𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))
5	1, 4	sylbi 120	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) → (𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))

Syntax hints:   → wi 4   ∨ wo 703   ∈ 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-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-ral 2453  df-rex 2454

This theorem is referenced by: (None)