Theorem 19.18 1046

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 148	. . . 4 ⊢ ((φ ↔ ψ) → (φ → ψ))
2	1	19.20i 989	. . 3 ⊢ (∀x(φ ↔ ψ) → ∀x(φ → ψ))
3		19.22 1035	. . 3 ⊢ (∀x(φ → ψ) → (∃xφ → ∃xψ))
4	2, 3	syl 10	. 2 ⊢ (∀x(φ ↔ ψ) → (∃xφ → ∃xψ))
5		bi2 149	. . . 4 ⊢ ((φ ↔ ψ) → (ψ → φ))
6	5	19.20i 989	. . 3 ⊢ (∀x(φ ↔ ψ) → ∀x(ψ → φ))
7		19.22 1035	. . 3 ⊢ (∀x(ψ → φ) → (∃xψ → ∃xφ))
8	6, 7	syl 10	. 2 ⊢ (∀x(φ ↔ ψ) → (∃xψ → ∃xφ))
9	4, 8	impbid 514	1 ⊢ (∀x(φ ↔ ψ) → (∃xφ ↔ ∃xψ))

Syntax hints:   → wi 3   ↔ wb 146  ∀wal 951  ∃wex 977

This theorem is referenced by:  exbii 1047  19.19 1051  exbid 1101  exintrbi 1114

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 960  ax-4 970  ax-5o 972

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978

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