Theorem spsd 2222

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 2218	. 2 ⊢ (∀𝑥𝜓 → 𝜓)
2		spsd.1	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
3	1, 2	syl5 34	1 ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))

Syntax hints:   → wi 4  ∀wal 1558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-12 2212

This theorem depends on definitions:  df-bi 209  df-ex 1800

This theorem is referenced by:  axc11v  2299  axc11rv  2300  equs5av  2311  equvel  2487  nfsb4t  2530  dfmoeu  2562  moexexlem  2653  2eu6  2683  zorn2lem4  10456  zorn2lem5  10457  axpowndlem3  10557  axacndlem5  10569  mh-setindnd  36894  axc11n11r  37155  wl-equsal1i  38044  axc5c4c711  44974