Theorem 2stdpc4 2354

Description: A double specialization using explicit substitution. This is Theorem PM*11.1 in [WhiteheadRussell] p. 159. See stdpc4 2353 for the analogous single specialization. See 2sp 2056 for another double specialization. (Contributed by Andrew Salmon, 24-May-2011.) (Revised by BJ, 21-Oct-2018.)

Assertion

Ref	Expression
2stdpc4	|- ( A. x A. y ph -> [ z / x ] [ w / y ] ph )

Proof of Theorem 2stdpc4

Step	Hyp	Ref	Expression
1		stdpc4 2353	. . 3 |- ( A. y ph -> [ w / y ] ph )
2	1	alimi 1739	. 2 |- ( A. x A. y ph -> A. x [ w / y ] ph )
3		stdpc4 2353	. 2 |- ( A. x [ w / y ] ph -> [ z / x ] [ w / y ] ph )
4	2, 3	syl 17	1 |- ( A. x A. y ph -> [ z / x ] [ w / y ] ph )

Syntax hints:   -> wi 4   A.wal 1481   [wsb 1880

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  df-sb 1881

This theorem is referenced by:  ax11-pm2  32823  pm11.11  38573