Theorem 19.36i 1872

Description: Inference from Theorem 19.36 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
19.36.1	⊢ Ⅎxψ
19.36i.2	⊢ ∃x(φ → ψ)

Assertion

Ref	Expression
19.36i	⊢ (∀xφ → ψ)

Proof of Theorem 19.36i

Step	Hyp	Ref	Expression
1		19.36i.2	. 2 ⊢ ∃x(φ → ψ)
2		19.36.1	. . 3 ⊢ Ⅎxψ
3	2	19.36 1871	. 2 ⊢ (∃x(φ → ψ) ↔ (∀xφ → ψ))
4	1, 3	mpbi 199	1 ⊢ (∀xφ → ψ)

Syntax hints:   → wi 4  ∀wal 1540  ∃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-an 360  df-ex 1542  df-nf 1545

This theorem is referenced by:  19.36aiv  1897  vtoclf  2909