Theorem 19.9h 1516

Description: A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of [Margaris] p. 89. (Contributed by FL, 24-Mar-2007.)

Hypothesis

Ref	Expression
19.9h.1	⊢ (φ → ∀xφ)

Assertion

Ref	Expression
19.9h	⊢ (∃xφ ↔ φ)

Proof of Theorem 19.9h

Step	Hyp	Ref	Expression
1		19.9ht 1514	. . 3 ⊢ (∀x(φ → ∀xφ) → (∃xφ → φ))
2		19.9h.1	. . 3 ⊢ (φ → ∀xφ)
3	1, 2	mpg 1320	. 2 ⊢ (∃xφ → φ)
4		19.8a 1464	. 2 ⊢ (φ → ∃xφ)
5	3, 4	impbii 117	1 ⊢ (∃xφ ↔ φ)

Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1226  ∃wex 1362

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-4 1381

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  19.9  1517  excomim  1535  exdistrfor  1663  sbcof2  1673  ax11ev  1691  19.9v  1733  exists1  1978