Theorem spimv 1990

Description: A version of spim 1975 with a distinct variable requirement instead of a bound variable hypothesis. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
spimv.1	⊢ (x = y → (φ → ψ))

Assertion

Ref	Expression
spimv	⊢ (∀xφ → ψ)

Distinct variable group:   ψ,x

Allowed substitution hints:   φ(x,y)   ψ(y)

Proof of Theorem spimv

Step	Hyp	Ref	Expression
1		nfv 1619	. 2 ⊢ Ⅎxψ
2		spimv.1	. 2 ⊢ (x = y → (φ → ψ))
3	1, 2	spim 1975	1 ⊢ (∀xφ → ψ)

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 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  aev  1991  spv  1998  ax16i  2046  aev-o  2182  reu6  3026