Theorem spsd 2184

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 2180	. 2 ⊢ (∀𝑥𝜓 → 𝜓)
2		spsd.1	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
3	1, 2	syl5 34	1 ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))

Syntax hints:   → wi 4  ∀wal 1536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2175

This theorem depends on definitions:  df-bi 210  df-ex 1782

This theorem is referenced by:  axc11v  2262  axc11rv  2263  equs5av  2276  equvel  2468  nfsb4t  2517  nfsb4tALT  2580  dfmoeu  2594  moexexlem  2688  2eu6  2719  zorn2lem4  9910  zorn2lem5  9911  axpowndlem3  10010  axacndlem5  10022  axc11n11r  34130  wl-equsal1i  34948  axc5c4c711  41105