Theorem r19.45mv 3309

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		r19.9rmv 3307	. . 3 ⊢ (∃x x ∈ A → (φ ↔ ∃x ∈ A φ))
2	1	orbi1d 704	. 2 ⊢ (∃x x ∈ A → ((φ ∨ ∃x ∈ A ψ) ↔ (∃x ∈ A φ ∨ ∃x ∈ A ψ)))
3		r19.43 2462	. 2 ⊢ (∃x ∈ A (φ ∨ ψ) ↔ (∃x ∈ A φ ∨ ∃x ∈ A ψ))
4	2, 3	syl6rbbr 188	1 ⊢ (∃x x ∈ A → (∃x ∈ A (φ ∨ ψ) ↔ (φ ∨ ∃x ∈ A ψ)))

Syntax hints:   → wi 4   ↔ wb 98   ∨ wo 628  ∃wex 1378   ∈ wcel 1390  ∃wrex 2301

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-cleq 2030  df-clel 2033  df-rex 2306

This theorem is referenced by:  ltexprlemloc  6581