Theorem 19.19 737

Description: Theorem 19.19 of [Margaris] p. 90.

Hypothesis

Ref	Expression
19.19.1	⊢ (φ → ∀xφ)

Assertion

Ref	Expression
19.19	⊢ (∀x(φ ↔ ψ) → (φ ↔ ∃xψ))

Proof of Theorem 19.19

Step	Hyp	Ref	Expression
1		19.18 732	. 2 ⊢ (∀x(φ ↔ ψ) → (∃xφ ↔ ∃xψ))
2		19.19.1	. . 3 ⊢ (φ → ∀xφ)
3	2	19.9r 718	. 2 ⊢ (φ ↔ ∃xφ)
4	1, 3	syl5bb 410	1 ⊢ (∀x(φ ↔ ψ) → (φ ↔ ∃xψ))

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672  ∃wex 678

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-an 198  df-ex 679

metamath.org