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Theorem sbccomlem 3116
 Description: Lemma for sbccom 3117. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 18-Oct-2016.)
Assertion
Ref Expression
sbccomlem ([̣A / x]̣[̣B / yφ ↔ [̣B / y]̣[̣A / xφ)
Distinct variable groups:   x,y,A   x,B,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem sbccomlem
StepHypRef Expression
1 excom 1741 . . . 4 (xy(x = A (y = B φ)) ↔ yx(x = A (y = B φ)))
2 exdistr 1906 . . . 4 (xy(x = A (y = B φ)) ↔ x(x = A y(y = B φ)))
3 an12 772 . . . . . . 7 ((x = A (y = B φ)) ↔ (y = B (x = A φ)))
43exbii 1582 . . . . . 6 (x(x = A (y = B φ)) ↔ x(y = B (x = A φ)))
5 19.42v 1905 . . . . . 6 (x(y = B (x = A φ)) ↔ (y = B x(x = A φ)))
64, 5bitri 240 . . . . 5 (x(x = A (y = B φ)) ↔ (y = B x(x = A φ)))
76exbii 1582 . . . 4 (yx(x = A (y = B φ)) ↔ y(y = B x(x = A φ)))
81, 2, 73bitr3i 266 . . 3 (x(x = A y(y = B φ)) ↔ y(y = B x(x = A φ)))
9 sbc5 3070 . . 3 ([̣A / xy(y = B φ) ↔ x(x = A y(y = B φ)))
10 sbc5 3070 . . 3 ([̣B / yx(x = A φ) ↔ y(y = B x(x = A φ)))
118, 9, 103bitr4i 268 . 2 ([̣A / xy(y = B φ) ↔ [̣B / yx(x = A φ))
12 sbc5 3070 . . 3 ([̣B / yφy(y = B φ))
1312sbcbii 3101 . 2 ([̣A / x]̣[̣B / yφ ↔ [̣A / xy(y = B φ))
14 sbc5 3070 . . 3 ([̣A / xφx(x = A φ))
1514sbcbii 3101 . 2 ([̣B / y]̣[̣A / xφ ↔ [̣B / yx(x = A φ))
1611, 13, 153bitr4i 268 1 ([̣A / x]̣[̣B / yφ ↔ [̣B / y]̣[̣A / xφ)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642  [̣wsbc 3046 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 2478  df-v 2861  df-sbc 3047 This theorem is referenced by:  sbccom  3117
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