Theorem 19.9h 1691

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 1689	. . 3 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → 𝜑))
2		19.9h.1	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3	1, 2	mpg 1499	. 2 ⊢ (∃𝑥𝜑 → 𝜑)
4		19.8a 1638	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
5	3, 4	impbii 126	1 ⊢ (∃𝑥𝜑 ↔ 𝜑)

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1395  ∃wex 1540

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 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  19.9  1692  excomim  1711  exdistrfor  1848  sbcof2  1858  ax11ev  1876  19.9v  1919  exists1  2176