Theorem 19.42vv 968

Description: Theorem 19.42 of [Margaris] p. 90 with 2 quantifiers.

Assertion

Ref	Expression
19.42vv	⊢ (∃x∃y(φ ∧ ψ) ↔ (φ ∧ ∃x∃yψ))

Distinct variable group(s):   φ,x   φ,y

Proof of Theorem 19.42vv

Step	Hyp	Ref	Expression
1		exdistr 967	. 2 ⊢ (∃x∃y(φ ∧ ψ) ↔ ∃x(φ ∧ ∃yψ))
2		19.42v 966	. 2 ⊢ (∃x(φ ∧ ∃yψ) ↔ (φ ∧ ∃x∃yψ))
3	1, 2	bitr 151	1 ⊢ (∃x∃y(φ ∧ ψ) ↔ (φ ∧ ∃x∃yψ))

Syntax hints:   ↔ wb 127   ∧ wa 196  ∃wex 678

This theorem is referenced by:  exdistr2 969  eeeanv 981  dfoprab2 3021  oprabex3 3046  oprabval3 3052  xpassen 3344  distrlem1pr 3921  distrlem5pr 3925

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

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