Theorem 19.9d 1782

Description: A deduction version of one direction of 19.9 1783. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)

Hypothesis

Ref	Expression
19.9d.1	⊢ (ψ → Ⅎxφ)

Assertion

Ref	Expression
19.9d	⊢ (ψ → (∃xφ → φ))

Proof of Theorem 19.9d

Step	Hyp	Ref	Expression
1		19.9d.1	. . 3 ⊢ (ψ → Ⅎxφ)
2		19.9t 1779	. . 3 ⊢ (Ⅎxφ → (∃xφ ↔ φ))
3	1, 2	syl 15	. 2 ⊢ (ψ → (∃xφ ↔ φ))
4	3	biimpd 198	1 ⊢ (ψ → (∃xφ → φ))

Syntax hints:   → wi 4   ↔ wb 176  ∃wex 1541  Ⅎwnf 1544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545

This theorem is referenced by:  exdistrf  1971  sbied  2036  sbequi  2059  copsexg  4608