Theorem equs5 1996

Description: Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
equs5	|- (-. A.x x = y -> (E.x(x = y /\ ph) -> A.x(x = y -> ph)))

Proof of Theorem equs5

Step	Hyp	Ref	Expression
1		nfnae 1956	. 2 |- F/x -. A.x x = y
2		nfa1 1788	. 2 |- F/xA.x(x = y -> ph)
3		ax11o 1994	. . 3 |- (-. A.x x = y -> (x = y -> (ph -> A.x(x = y -> ph))))
4	3	imp3a 420	. 2 |- (-. A.x x = y -> ((x = y /\ ph) -> A.x(x = y -> ph)))
5	1, 2, 4	exlimd 1806	1 |- (-. A.x x = y -> (E.x(x = y /\ ph) -> A.x(x = y -> ph)))

Syntax hints:  -. wn 3   -> wi 4   /\ wa 358  A.wal 1540  E.wex 1541

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:  sb3  2052  sb4  2053