Theorem eu1 2225

Description: An alternate way to express uniqueness used by some authors. Exercise 2(b) of [Margaris] p. 110. (Contributed by NM, 20-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)

Hypothesis

Ref	Expression
eu1.1	|- F/yph

Assertion

Ref	Expression
eu1	|- (E!xph <-> E.x(ph /\ A.y([y / x]ph -> x = y)))

Distinct variable group:   x,y

Allowed substitution hints:   ph(x,y)

Proof of Theorem eu1

Step	Hyp	Ref	Expression
1		nfs1v 2106	. . 3 |- F/x[y / x]ph
2	1	euf 2210	. 2 |- (E!y[y / x]ph <-> E.xA.y([y / x]ph <-> y = x))
3		eu1.1	. . 3 |- F/yph
4	3	sb8eu 2222	. 2 |- (E!xph <-> E!y[y / x]ph)
5		equcom 1680	. . . . . . 7 |- (x = y <-> y = x)
6	5	imbi2i 303	. . . . . 6 |- (([y / x]ph -> x = y) <-> ([y / x]ph -> y = x))
7	6	albii 1566	. . . . 5 |- (A.y([y / x]ph -> x = y) <-> A.y([y / x]ph -> y = x))
8	3	sb6rf 2091	. . . . 5 |- (ph <-> A.y(y = x -> [y / x]ph))
9	7, 8	anbi12i 678	. . . 4 |- ((A.y([y / x]ph -> x = y) /\ ph) <-> (A.y([y / x]ph -> y = x) /\ A.y(y = x -> [y / x]ph)))
10		ancom 437	. . . 4 |- ((ph /\ A.y([y / x]ph -> x = y)) <-> (A.y([y / x]ph -> x = y) /\ ph))
11		albiim 1611	. . . 4 |- (A.y([y / x]ph <-> y = x) <-> (A.y([y / x]ph -> y = x) /\ A.y(y = x -> [y / x]ph)))
12	9, 10, 11	3bitr4i 268	. . 3 |- ((ph /\ A.y([y / x]ph -> x = y)) <-> A.y([y / x]ph <-> y = x))
13	12	exbii 1582	. 2 |- (E.x(ph /\ A.y([y / x]ph -> x = y)) <-> E.xA.y([y / x]ph <-> y = x))
14	2, 4, 13	3bitr4i 268	1 |- (E!xph <-> E.x(ph /\ A.y([y / x]ph -> x = y)))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541   F/wnf 1544   = wceq 1642  [wsb 1648  E!weu 2204

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-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208

This theorem is referenced by:  euex  2227  eu2  2229  fvfullfunlem1  5862