Theorem spimv 2257

Description: A version of spim 2254 with a distinct variable requirement instead of a bound variable hypothesis. See also spimv1 2115 and spimvw 1927. See also spimvALT 2258. (Contributed by NM, 31-Jul-1993.) Removed dependency on ax-10 2019. (Revised by BJ, 29-Nov-2020.)

Hypothesis

Ref	Expression
spimv.1	|- ( x = y -> ( ph -> ps ) )

Assertion

Ref	Expression
spimv	|- ( A. x ph -> ps )

Distinct variable group:   ps, x

Allowed substitution hints:   ph( x, y)   ps( y)

Proof of Theorem spimv

Step	Hyp	Ref	Expression
1		ax6e 2250	. . 3 |- E. x x = y
2		spimv.1	. . 3 |- ( x = y -> ( ph -> ps ) )
3	1, 2	eximii 1764	. 2 |- E. x ( ph -> ps )
4	3	19.36iv 1905	1 |- ( A. x ph -> ps )

Syntax hints:   -> wi 4   A.wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by:  spv  2260  aevALTOLD  2321  axc16i  2322  reu6  3395  el  4847  aev-o  34216  axc11next  38607