Theorem spsd 2186

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 2182	. 2 ⊢ (∀𝑥𝜓 → 𝜓)
2		spsd.1	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
3	1, 2	syl5 34	1 ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))

Syntax hints:   → wi 4  ∀wal 1535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-ex 1781

This theorem is referenced by:  axc11v  2265  axc11rv  2266  equs5av  2279  equvel  2479  nfsb4t  2539  nfsb4tALT  2604  dfmoeu  2618  moexexlem  2711  2eu6  2742  zorn2lem4  9923  zorn2lem5  9924  axpowndlem3  10023  axacndlem5  10035  axc11n11r  34019  wl-equsal1i  34785  axc5c4c711  40740