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Theorem strcoll2 7397
Description: Version of ax-strcoll 7396 with one DV condition removed and without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.)
Assertion
Ref Expression
strcoll2 (x 𝑎 yφ𝑏y(y 𝑏x 𝑎 φ))
Distinct variable groups:   𝑎,𝑏,x,y   φ,𝑏
Allowed substitution hints:   φ(x,y,𝑎)

Proof of Theorem strcoll2
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 raleq 2479 . . 3 (z = 𝑎 → (x z yφx 𝑎 yφ))
2 rexeq 2480 . . . . . 6 (z = 𝑎 → (x z φx 𝑎 φ))
32bibi2d 221 . . . . 5 (z = 𝑎 → ((y 𝑏x z φ) ↔ (y 𝑏x 𝑎 φ)))
43albidv 1683 . . . 4 (z = 𝑎 → (y(y 𝑏x z φ) ↔ y(y 𝑏x 𝑎 φ)))
54exbidv 1684 . . 3 (z = 𝑎 → (𝑏y(y 𝑏x z φ) ↔ 𝑏y(y 𝑏x 𝑎 φ)))
61, 5imbi12d 223 . 2 (z = 𝑎 → ((x z yφ𝑏y(y 𝑏x z φ)) ↔ (x 𝑎 yφ𝑏y(y 𝑏x 𝑎 φ))))
7 ax-strcoll 7396 . . 3 z(x z yφ𝑏y(y 𝑏x z φ))
87spi 1407 . 2 (x z yφ𝑏y(y 𝑏x z φ))
96, 8chvarv 1790 1 (x 𝑎 yφ𝑏y(y 𝑏x 𝑎 φ))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1224  wex 1358  wral 2280  wrex 2281
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-strcoll 7396
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286
This theorem is referenced by:  strcollnft  7398  strcollnfALT  7400
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