Theorem exlimd 1806

Description: Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)

Hypotheses

Ref	Expression
exlimd.1	⊢ Ⅎxφ
exlimd.2	⊢ Ⅎxχ
exlimd.3	⊢ (φ → (ψ → χ))

Assertion

Ref	Expression
exlimd	⊢ (φ → (∃xψ → χ))

Proof of Theorem exlimd

Step	Hyp	Ref	Expression
1		exlimd.1	. . 3 ⊢ Ⅎxφ
2		exlimd.3	. . 3 ⊢ (φ → (ψ → χ))
3	1, 2	alrimi 1765	. 2 ⊢ (φ → ∀x(ψ → χ))
4		exlimd.2	. . 3 ⊢ Ⅎxχ
5	4	19.23 1801	. 2 ⊢ (∀x(ψ → χ) ↔ (∃xψ → χ))
6	3, 5	sylib 188	1 ⊢ (φ → (∃xψ → χ))

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-ex 1542  df-nf 1545

This theorem is referenced by:  exlimdh  1807  exlimdd  1889  equs5  1996  exists2  2294  ceqsalg  2883  copsex2t  4608  mosubopt  4612  ov3  5599