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Theorem strcoll2 9367
Description: Version of ax-strcoll 9366 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 2499 . . 3 (z = 𝑎 → (x z yφx 𝑎 yφ))
2 rexeq 2500 . . . . . 6 (z = 𝑎 → (x z φx 𝑎 φ))
32bibi2d 221 . . . . 5 (z = 𝑎 → ((y 𝑏x z φ) ↔ (y 𝑏x 𝑎 φ)))
43albidv 1702 . . . 4 (z = 𝑎 → (y(y 𝑏x z φ) ↔ y(y 𝑏x 𝑎 φ)))
54exbidv 1703 . . 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 9366 . . 3 z(x z yφ𝑏y(y 𝑏x z φ))
87spi 1426 . 2 (x z yφ𝑏y(y 𝑏x z φ))
96, 8chvarv 1809 1 (x 𝑎 yφ𝑏y(y 𝑏x 𝑎 φ))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1240  wex 1378  wral 2300  wrex 2301
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  ax-strcoll 9366
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306
This theorem is referenced by:  strcollnft  9368  strcollnfALT  9370
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