Theorem spsd 2178

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 2174	. 2 ⊢ (∀𝑥𝜓 → 𝜓)
2		spsd.1	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
3	1, 2	syl5 34	1 ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))

Syntax hints:   → wi 4  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-12 2169

This theorem depends on definitions:  df-bi 206  df-ex 1780

This theorem is referenced by:  axc11v  2254  axc11rv  2255  equs5av  2269  equvel  2454  nfsb4t  2501  dfmoeu  2534  moexexlem  2626  2eu6  2656  ab0OLD  4315  zorn2lem4  10301  zorn2lem5  10302  axpowndlem3  10401  axacndlem5  10413  axc11n11r  34910  wl-equsal1i  35746  axc5c4c711  42057