Theorem spsd 2181

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 2177	. 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 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-12 2172

This theorem depends on definitions:  df-bi 206  df-ex 1783

This theorem is referenced by:  axc11v  2256  axc11rv  2257  equs5av  2271  equvel  2456  nfsb4t  2499  dfmoeu  2531  moexexlem  2623  2eu6  2653  ab0OLD  4375  zorn2lem4  10491  zorn2lem5  10492  axpowndlem3  10591  axacndlem5  10603  axc11n11r  35550  wl-equsal1i  36401  axc5c4c711  43146