Theorem 19.41 774

Description: Theorem 19.41 of [Margaris] p. 90.

Hypothesis

Ref	Expression
19.41.1	⊢ (ψ → ∀xψ)

Assertion

Ref	Expression
19.41	⊢ (∃x(φ ∧ ψ) ↔ (∃xφ ∧ ψ))

Proof of Theorem 19.41

Step	Hyp	Ref	Expression
1		df-ex 679	. 2 ⊢ (∃x(φ ∧ ψ) ↔ ¬ ∀x ¬ (φ ∧ ψ))
2		19.41.1	. . . . . 6 ⊢ (ψ → ∀xψ)
3	2	hbne 699	. . . . 5 ⊢ (¬ ψ → ∀x ¬ ψ)
4	3	19.31 766	. . . 4 ⊢ (∀x(¬ φ ∨ ¬ ψ) ↔ (∀x ¬ φ ∨ ¬ ψ))
5		ianor 253	. . . . 5 ⊢ (¬ (φ ∧ ψ) ↔ (¬ φ ∨ ¬ ψ))
6	5	bial 695	. . . 4 ⊢ (∀x ¬ (φ ∧ ψ) ↔ ∀x(¬ φ ∨ ¬ ψ))
7		ianor 253	. . . . 5 ⊢ (¬ (∃xφ ∧ ψ) ↔ (¬ ∃xφ ∨ ¬ ψ))
8		alnex 716	. . . . . 6 ⊢ (∀x ¬ φ ↔ ¬ ∃xφ)
9	8	orbi1i 215	. . . . 5 ⊢ ((∀x ¬ φ ∨ ¬ ψ) ↔ (¬ ∃xφ ∨ ¬ ψ))
10	7, 9	bitr4 154	. . . 4 ⊢ (¬ (∃xφ ∧ ψ) ↔ (∀x ¬ φ ∨ ¬ ψ))
11	4, 6, 10	3bitr4 158	. . 3 ⊢ (∀x ¬ (φ ∧ ψ) ↔ ¬ (∃xφ ∧ ψ))
12	11	bicon2i 194	. 2 ⊢ ((∃xφ ∧ ψ) ↔ ¬ ∀x ¬ (φ ∧ ψ))
13	1, 12	bitr4 154	1 ⊢ (∃x(φ ∧ ψ) ↔ (∃xφ ∧ ψ))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196  ∀wal 672  ∃wex 678

This theorem is referenced by:  19.42 775  sbf 870  hbs1f 874  19.41v 963  eeanv 980  2euex 1061  2exeu 1066

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

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

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