Theorem 19.15 694

Description: Theorem 19.15 of [Margaris] p. 90.

Assertion

Ref	Expression
19.15	⊢ (∀x(φ ↔ ψ) → (∀xφ ↔ ∀xψ))

Proof of Theorem 19.15

Step	Hyp	Ref	Expression
1		bi1 130	. . 3 ⊢ ((φ ↔ ψ) → (φ → ψ))
2	1	19.20ii 692	. 2 ⊢ (∀x(φ ↔ ψ) → (∀xφ → ∀xψ))
3		bi2 131	. . 3 ⊢ ((φ ↔ ψ) → (ψ → φ))
4	3	19.20ii 692	. 2 ⊢ (∀x(φ ↔ ψ) → (∀xψ → ∀xφ))
5	2, 4	impbid 397	1 ⊢ (∀x(φ ↔ ψ) → (∀xφ ↔ ∀xψ))

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

This theorem is referenced by:  bial 695  19.16 730  19.17 731  19.33b 771  biald 782  sbal1 996

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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