Theorem r19.42v 1303

Description: Restricted version of Theorem 19.42 of [Margaris] p. 90.

Assertion

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

Distinct variable group(s):   φ,x

Proof of Theorem r19.42v

Step	Hyp	Ref	Expression
1		r19.41v 1302	. 2 ⊢ (∃x ∈ A (ψ ∧ φ) ↔ (∃x ∈ A ψ ∧ φ))
2		ancom 333	. . 3 ⊢ ((φ ∧ ψ) ↔ (ψ ∧ φ))
3	2	birex 1224	. 2 ⊢ (∃x ∈ A (φ ∧ ψ) ↔ ∃x ∈ A (ψ ∧ φ))
4		ancom 333	. 2 ⊢ ((φ ∧ ∃x ∈ A ψ) ↔ (∃x ∈ A ψ ∧ φ))
5	1, 3, 4	3bitr4 158	1 ⊢ (∃x ∈ A (φ ∧ ψ) ↔ (φ ∧ ∃x ∈ A ψ))

Syntax hints:   ↔ wb 127   ∧ wa 196  ∃wrex 1202

This theorem is referenced by:  2reuswap 1341  iunrab 2022  iunin2 2030  iundif2 2032  elxp2 2443  cnvuni 2521  f1oiso 2942  tfrlem8 2956  trcl 3489  aceq5lem2 3559  sumdmdi 5785

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-17 925

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-rex 1206

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