Theorem 19.41vv 1902

Description: Theorem 19.41 of [Margaris] p. 90 with 2 quantifiers. (Contributed by NM, 30-Apr-1995.)

Assertion

Ref	Expression
19.41vv	⊢ (∃x∃y(φ ∧ ψ) ↔ (∃x∃yφ ∧ ψ))

Distinct variable groups:   ψ,x   ψ,y

Allowed substitution hints:   φ(x,y)

Proof of Theorem 19.41vv

Step	Hyp	Ref	Expression
1		19.41v 1901	. . 3 ⊢ (∃y(φ ∧ ψ) ↔ (∃yφ ∧ ψ))
2	1	exbii 1582	. 2 ⊢ (∃x∃y(φ ∧ ψ) ↔ ∃x(∃yφ ∧ ψ))
3		19.41v 1901	. 2 ⊢ (∃x(∃yφ ∧ ψ) ↔ (∃x∃yφ ∧ ψ))
4	2, 3	bitri 240	1 ⊢ (∃x∃y(φ ∧ ψ) ↔ (∃x∃yφ ∧ ψ))

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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

This theorem is referenced by:  19.41vvv  1903  pm11.07  2115  dfpw12  4301  ltfinex  4464  setconslem4  4734  setconslem6  4736  rabxp  4814  elres  4995  fnov  5591  mpt2mptx  5708  restxp  5786  lecex  6115