Theorem spsd 2188

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 2184	. 2 ⊢ (∀𝑥𝜓 → 𝜓)
2		spsd.1	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
3	1, 2	syl5 34	1 ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))

Syntax hints:   → wi 4  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-12 2179

This theorem depends on definitions:  df-bi 210  df-ex 1787

This theorem is referenced by:  axc11v  2265  axc11rv  2266  equs5av  2279  equvel  2457  nfsb4t  2504  dfmoeu  2537  moexexlem  2630  2eu6  2660  ab0OLD  4274  zorn2lem4  10011  zorn2lem5  10012  axpowndlem3  10111  axacndlem5  10123  axc11n11r  34520  wl-equsal1i  35357  axc5c4c711  41597