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Theorem sbccomlem 2826
Description: Lemma for sbccom 2827. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 18-Oct-2016.)
Assertion
Ref Expression
sbccomlem ([A / x][B / y]φ[B / y][A / x]φ)
Distinct variable groups:   x,y,A   x,B,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem sbccomlem
StepHypRef Expression
1 excom 1551 . . . 4 (xy(x = A (y = B φ)) ↔ yx(x = A (y = B φ)))
2 exdistr 1784 . . . 4 (xy(x = A (y = B φ)) ↔ x(x = A y(y = B φ)))
3 an12 495 . . . . . . 7 ((x = A (y = B φ)) ↔ (y = B (x = A φ)))
43exbii 1493 . . . . . 6 (x(x = A (y = B φ)) ↔ x(y = B (x = A φ)))
5 19.42v 1783 . . . . . 6 (x(y = B (x = A φ)) ↔ (y = B x(x = A φ)))
64, 5bitri 173 . . . . 5 (x(x = A (y = B φ)) ↔ (y = B x(x = A φ)))
76exbii 1493 . . . 4 (yx(x = A (y = B φ)) ↔ y(y = B x(x = A φ)))
81, 2, 73bitr3i 199 . . 3 (x(x = A y(y = B φ)) ↔ y(y = B x(x = A φ)))
9 sbc5 2781 . . 3 ([A / x]y(y = B φ) ↔ x(x = A y(y = B φ)))
10 sbc5 2781 . . 3 ([B / y]x(x = A φ) ↔ y(y = B x(x = A φ)))
118, 9, 103bitr4i 201 . 2 ([A / x]y(y = B φ) ↔ [B / y]x(x = A φ))
12 sbc5 2781 . . 3 ([B / y]φy(y = B φ))
1312sbcbii 2812 . 2 ([A / x][B / y]φ[A / x]y(y = B φ))
14 sbc5 2781 . . 3 ([A / x]φx(x = A φ))
1514sbcbii 2812 . 2 ([B / y][A / x]φ[B / y]x(x = A φ))
1611, 13, 153bitr4i 201 1 ([A / x][B / y]φ[B / y][A / x]φ)
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1242  wex 1378  [wsbc 2758
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759
This theorem is referenced by:  sbccom  2827
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