Theorem r19.44mv 3452

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

Assertion

Ref	Expression
r19.44mv	⊢ (∃𝑦 𝑦 ∈ 𝐴 → (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ 𝜓)))

Distinct variable groups:   𝑥,𝐴   𝑦,𝐴   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)

Proof of Theorem r19.44mv

Step	Hyp	Ref	Expression
1		r19.9rmv 3449	. . 3 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜓))
2	1	orbi2d 779	. 2 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → ((∃𝑥 ∈ 𝐴 𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓)))
3		r19.43 2587	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))
4	2, 3	syl6rbbr 198	1 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ 𝜓)))

Syntax hints:   → wi 4   ↔ wb 104   ∨ wo 697  ∃wex 1468   ∈ wcel 1480  ∃wrex 2415

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-cleq 2130  df-clel 2133  df-rex 2420

This theorem is referenced by:  frecabcl  6289