Theorem r19.42v 2766

Description: Restricted version of Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)

Assertion

Ref	Expression
r19.42v	⊢ (∃x ∈ A (φ ∧ ψ) ↔ (φ ∧ ∃x ∈ A ψ))

Distinct variable group:   φ,x

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

Proof of Theorem r19.42v

Step	Hyp	Ref	Expression
1		r19.41v 2765	. 2 ⊢ (∃x ∈ A (ψ ∧ φ) ↔ (∃x ∈ A ψ ∧ φ))
2		ancom 437	. . 3 ⊢ ((φ ∧ ψ) ↔ (ψ ∧ φ))
3	2	rexbii 2640	. 2 ⊢ (∃x ∈ A (φ ∧ ψ) ↔ ∃x ∈ A (ψ ∧ φ))
4		ancom 437	. 2 ⊢ ((φ ∧ ∃x ∈ A ψ) ↔ (∃x ∈ A ψ ∧ φ))
5	1, 3, 4	3bitr4i 268	1 ⊢ (∃x ∈ A (φ ∧ ψ) ↔ (φ ∧ ∃x ∈ A ψ))

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃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-tru 1319  df-ex 1542  df-nf 1545  df-rex 2621

This theorem is referenced by:  ceqsrexbv  2974  ceqsrex2v  2975  2reuswap  3039  2reu5  3045  iunrab  4014  iunin2  4031  iundif2  4034  addcass  4416  elxp2  4803  cnvuni  4896  f1oiso  5500