Theorem 19.18 732

Description: Theorem 19.18 of [Margaris] p. 90.

Assertion

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

Proof of Theorem 19.18

Step	Hyp	Ref	Expression
1		bi1 130	. . . 4 ⊢ ((φ ↔ ψ) → (φ → ψ))
2	1	19.20i 691	. . 3 ⊢ (∀x(φ ↔ ψ) → ∀x(φ → ψ))
3		19.22 722	. . 3 ⊢ (∀x(φ → ψ) → (∃xφ → ∃xψ))
4	2, 3	syl 12	. 2 ⊢ (∀x(φ ↔ ψ) → (∃xφ → ∃xψ))
5		bi2 131	. . . 4 ⊢ ((φ ↔ ψ) → (ψ → φ))
6	5	19.20i 691	. . 3 ⊢ (∀x(φ ↔ ψ) → ∀x(ψ → φ))
7		19.22 722	. . 3 ⊢ (∀x(ψ → φ) → (∃xψ → ∃xφ))
8	6, 7	syl 12	. 2 ⊢ (∀x(φ ↔ ψ) → (∃xψ → ∃xφ))
9	4, 8	impbid 397	1 ⊢ (∀x(φ ↔ ψ) → (∃xφ ↔ ∃xψ))

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

This theorem is referenced by:  biex 733  19.19 737  biexd 783

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  df-ex 679

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