Theorem 19.41vv 964

Description: Theorem 19.41 of [Margaris] p. 90 with 2 quantifiers.

Assertion

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

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

Proof of Theorem 19.41vv

Step	Hyp	Ref	Expression
1		19.41v 963	. . 3 ⊢ (∃y(φ ∧ ψ) ↔ (∃yφ ∧ ψ))
2	1	biex 733	. 2 ⊢ (∃x∃y(φ ∧ ψ) ↔ ∃x(∃yφ ∧ ψ))
3		19.41v 963	. 2 ⊢ (∃x(∃yφ ∧ ψ) ↔ (∃x∃yφ ∧ ψ))
4	2, 3	bitr 151	1 ⊢ (∃x∃y(φ ∧ ψ) ↔ (∃x∃yφ ∧ ψ))

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

This theorem is referenced by:  19.41vvv 965  cbvop 2473  fnoprval 3042  xpcomen 3343  xpassen 3344  aceq5lem1 3558  genpass 3906  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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