Theorem 19.27 750

Description: Theorem 19.27 of [Margaris] p. 90.

Hypothesis

Ref	Expression
19.27.1	⊢ (ψ → ∀xψ)

Assertion

Ref	Expression
19.27	⊢ (∀x(φ ∧ ψ) ↔ (∀xφ ∧ ψ))

Proof of Theorem 19.27

Step	Hyp	Ref	Expression
1		19.26 749	. 2 ⊢ (∀x(φ ∧ ψ) ↔ (∀xφ ∧ ∀xψ))
2		19.27.1	. . . 4 ⊢ (ψ → ∀xψ)
3	2	19.3r 714	. . 3 ⊢ (ψ ↔ ∀xψ)
4	3	anbi2i 367	. 2 ⊢ ((∀xφ ∧ ψ) ↔ (∀xφ ∧ ∀xψ))
5	1, 4	bitr4 154	1 ⊢ (∀x(φ ∧ ψ) ↔ (∀xφ ∧ ψ))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672

This theorem is referenced by:  exan 784  aaan 794  19.27v 956

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-gen 677

This theorem depends on definitions:  df-bi 128  df-an 198

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