Theorem 19.9h 1657

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 1655	. . 3 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → 𝜑))
2		19.9h.1	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3	1, 2	mpg 1465	. 2 ⊢ (∃𝑥𝜑 → 𝜑)
4		19.8a 1604	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
5	3, 4	impbii 126	1 ⊢ (∃𝑥𝜑 ↔ 𝜑)

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1362  ∃wex 1506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  19.9  1658  excomim  1677  exdistrfor  1814  sbcof2  1824  ax11ev  1842  19.9v  1885  exists1  2141