Theorem r19.41 2764

Description: Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 1-Nov-2010.)

Hypothesis

Ref	Expression
r19.41.1	⊢ Ⅎxψ

Assertion

Ref	Expression
r19.41	⊢ (∃x ∈ A (φ ∧ ψ) ↔ (∃x ∈ A φ ∧ ψ))

Proof of Theorem r19.41

Step	Hyp	Ref	Expression
1		anass 630	. . . 4 ⊢ (((x ∈ A ∧ φ) ∧ ψ) ↔ (x ∈ A ∧ (φ ∧ ψ)))
2	1	exbii 1582	. . 3 ⊢ (∃x((x ∈ A ∧ φ) ∧ ψ) ↔ ∃x(x ∈ A ∧ (φ ∧ ψ)))
3		r19.41.1	. . . 4 ⊢ Ⅎxψ
4	3	19.41 1879	. . 3 ⊢ (∃x((x ∈ A ∧ φ) ∧ ψ) ↔ (∃x(x ∈ A ∧ φ) ∧ ψ))
5	2, 4	bitr3i 242	. 2 ⊢ (∃x(x ∈ A ∧ (φ ∧ ψ)) ↔ (∃x(x ∈ A ∧ φ) ∧ ψ))
6		df-rex 2621	. 2 ⊢ (∃x ∈ A (φ ∧ ψ) ↔ ∃x(x ∈ A ∧ (φ ∧ ψ)))
7		df-rex 2621	. . 3 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ∧ φ))
8	7	anbi1i 676	. 2 ⊢ ((∃x ∈ A φ ∧ ψ) ↔ (∃x(x ∈ A ∧ φ) ∧ ψ))
9	5, 6, 8	3bitr4i 268	1 ⊢ (∃x ∈ A (φ ∧ ψ) ↔ (∃x ∈ A φ ∧ ψ))

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541  Ⅎwnf 1544   ∈ wcel 1710  ∃wrex 2616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-rex 2621

This theorem is referenced by:  r19.41v  2765