Theorem cbv3hv 2174

Description: Version of cbv3h 2266 with a dv condition on x , y, which does not require ax-13 2246. Was used in a proof of axc11n 2307 (but of independent interest). (Contributed by NM, 25-Jul-2015.) (Proof shortened by Wolf Lammen, 29-Nov-2020.) (Proof shortened by BJ, 30-Nov-2020.)

Hypotheses

Ref	Expression
cbv3hv.nf1	|- ( ph -> A. y ph )
cbv3hv.nf2	|- ( ps -> A. x ps )
cbv3hv.1	|- ( x = y -> ( ph -> ps ) )

Assertion

Ref	Expression
cbv3hv	|- ( A. x ph -> A. y ps )

Distinct variable group:   x, y

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

Proof of Theorem cbv3hv

Step	Hyp	Ref	Expression
1		cbv3hv.nf1	. . 3 |- ( ph -> A. y ph )
2	1	nf5i 2024	. 2 |- F/ y ph
3		cbv3hv.nf2	. . 3 |- ( ps -> A. x ps )
4	3	nf5i 2024	. 2 |- F/ x ps
5		cbv3hv.1	. 2 |- ( x = y -> ( ph -> ps ) )
6	2, 4, 5	cbv3v 2172	1 |- ( A. x ph -> A. y 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-10 2019  ax-11 2034  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)