Theorem spsd 2223

Description: Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.)

Hypothesis

Ref	Expression
spsd.1	⊢ (𝜑 → (𝜓 → 𝜒))

Assertion

Ref	Expression
spsd	⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))

Proof of Theorem spsd

Step	Hyp	Ref	Expression
1		sp 2219	. 2 ⊢ (∀𝑥𝜓 → 𝜓)
2		spsd.1	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
3	1, 2	syl5 35	1 ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))

This proof depends on syntax axioms:   → wi 4  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-12 2213

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by:  axc11v  2298  axc11rv  2299  equs5av  2310  equvel  2485  nfsb4t  2528  dfmoeu  2560  moexexlem  2651  2eu6  2681  zorn2lem4  10501  zorn2lem5  10502  axpowndlem3  10608  axacndlem5  10620  mh-setindnd  37156  axc11n11r  37416  wl-equsal1i  38307  axc5c4c711  45225