Theorem sbccomlem 3117

Description: Lemma for sbccom 3118. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 18-Oct-2016.)

Assertion

Ref	Expression
sbccomlem	|- ( [.A / x ]. [.B / y ].ph <-> [.B / y ]. [.A / x ].ph)

Distinct variable groups:   x,y,A   x,B,y

Allowed substitution hints:   ph(x,y)

Proof of Theorem sbccomlem

Step	Hyp	Ref	Expression
1		excom 1741	. . . 4 |- (E.xE.y(x = A /\ (y = B /\ ph)) <-> E.yE.x(x = A /\ (y = B /\ ph)))
2		exdistr 1906	. . . 4 |- (E.xE.y(x = A /\ (y = B /\ ph)) <-> E.x(x = A /\ E.y(y = B /\ ph)))
3		an12 772	. . . . . . 7 |- ((x = A /\ (y = B /\ ph)) <-> (y = B /\ (x = A /\ ph)))
4	3	exbii 1582	. . . . . 6 |- (E.x(x = A /\ (y = B /\ ph)) <-> E.x(y = B /\ (x = A /\ ph)))
5		19.42v 1905	. . . . . 6 |- (E.x(y = B /\ (x = A /\ ph)) <-> (y = B /\ E.x(x = A /\ ph)))
6	4, 5	bitri 240	. . . . 5 |- (E.x(x = A /\ (y = B /\ ph)) <-> (y = B /\ E.x(x = A /\ ph)))
7	6	exbii 1582	. . . 4 |- (E.yE.x(x = A /\ (y = B /\ ph)) <-> E.y(y = B /\ E.x(x = A /\ ph)))
8	1, 2, 7	3bitr3i 266	. . 3 |- (E.x(x = A /\ E.y(y = B /\ ph)) <-> E.y(y = B /\ E.x(x = A /\ ph)))
9		sbc5 3071	. . 3 |- ( [.A / x ].E.y(y = B /\ ph) <-> E.x(x = A /\ E.y(y = B /\ ph)))
10		sbc5 3071	. . 3 |- ( [.B / y ].E.x(x = A /\ ph) <-> E.y(y = B /\ E.x(x = A /\ ph)))
11	8, 9, 10	3bitr4i 268	. 2 |- ( [.A / x ].E.y(y = B /\ ph) <-> [.B / y ].E.x(x = A /\ ph))
12		sbc5 3071	. . 3 |- ( [.B / y ].ph <-> E.y(y = B /\ ph))
13	12	sbcbii 3102	. 2 |- ( [.A / x ]. [.B / y ].ph <-> [.A / x ].E.y(y = B /\ ph))
14		sbc5 3071	. . 3 |- ( [.A / x ].ph <-> E.x(x = A /\ ph))
15	14	sbcbii 3102	. 2 |- ( [.B / y ]. [.A / x ].ph <-> [.B / y ].E.x(x = A /\ ph))
16	11, 13, 15	3bitr4i 268	1 |- ( [.A / x ]. [.B / y ].ph <-> [.B / y ]. [.A / x ].ph)

Syntax hints:   <-> wb 176   /\ wa 358  E.wex 1541   = wceq 1642   [.wsbc 3047

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  ax-ext 2334

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-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048

This theorem is referenced by:  sbccom  3118