Theorem r19.44mv 3360

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 3357	. . 3 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜓))
2	1	orbi2d 737	. 2 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → ((∃𝑥 ∈ 𝐴 𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓)))
3		r19.43 2520	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))
4	2, 3	syl6rbbr 197	1 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ 𝜓)))

Syntax hints:   → wi 4   ↔ wb 103   ∨ wo 662  ∃wex 1424   ∈ wcel 1436  ∃wrex 2356

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-ext 2067

This theorem depends on definitions:  df-bi 115  df-cleq 2078  df-clel 2081  df-rex 2361

This theorem is referenced by:  frecabcl  6099