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Theorem sscoll2 9418
Description: Version of ax-sscoll 9417 with two DV conditions removed and without initial universal quantifiers. (Contributed by BJ, 5-Oct-2019.)
Assertion
Ref Expression
sscoll2 𝑐z(x 𝑎 y 𝑏 φ𝑑 𝑐 y(y 𝑑x 𝑎 φ))
Distinct variable groups:   𝑎,𝑏,𝑐,𝑑,x,y,z   φ,𝑐,𝑑
Allowed substitution hints:   φ(x,y,z,𝑎,𝑏)

Proof of Theorem sscoll2
Dummy variables u v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1418 . . 3 𝑐(u = 𝑎 v = 𝑏)
2 nfv 1418 . . . 4 z(u = 𝑎 v = 𝑏)
3 simpl 102 . . . . . 6 ((u = 𝑎 v = 𝑏) → u = 𝑎)
4 rexeq 2500 . . . . . . 7 (v = 𝑏 → (y v φy 𝑏 φ))
54adantl 262 . . . . . 6 ((u = 𝑎 v = 𝑏) → (y v φy 𝑏 φ))
63, 5raleqbidv 2511 . . . . 5 ((u = 𝑎 v = 𝑏) → (x u y v φx 𝑎 y 𝑏 φ))
7 nfv 1418 . . . . . 6 𝑑(u = 𝑎 v = 𝑏)
8 nfv 1418 . . . . . . 7 y(u = 𝑎 v = 𝑏)
9 rexeq 2500 . . . . . . . . 9 (u = 𝑎 → (x u φx 𝑎 φ))
109adantr 261 . . . . . . . 8 ((u = 𝑎 v = 𝑏) → (x u φx 𝑎 φ))
1110bibi2d 221 . . . . . . 7 ((u = 𝑎 v = 𝑏) → ((y 𝑑x u φ) ↔ (y 𝑑x 𝑎 φ)))
128, 11albid 1503 . . . . . 6 ((u = 𝑎 v = 𝑏) → (y(y 𝑑x u φ) ↔ y(y 𝑑x 𝑎 φ)))
137, 12rexbid 2319 . . . . 5 ((u = 𝑎 v = 𝑏) → (𝑑 𝑐 y(y 𝑑x u φ) ↔ 𝑑 𝑐 y(y 𝑑x 𝑎 φ)))
146, 13imbi12d 223 . . . 4 ((u = 𝑎 v = 𝑏) → ((x u y v φ𝑑 𝑐 y(y 𝑑x u φ)) ↔ (x 𝑎 y 𝑏 φ𝑑 𝑐 y(y 𝑑x 𝑎 φ))))
152, 14albid 1503 . . 3 ((u = 𝑎 v = 𝑏) → (z(x u y v φ𝑑 𝑐 y(y 𝑑x u φ)) ↔ z(x 𝑎 y 𝑏 φ𝑑 𝑐 y(y 𝑑x 𝑎 φ))))
161, 15exbid 1504 . 2 ((u = 𝑎 v = 𝑏) → (𝑐z(x u y v φ𝑑 𝑐 y(y 𝑑x u φ)) ↔ 𝑐z(x 𝑎 y 𝑏 φ𝑑 𝑐 y(y 𝑑x 𝑎 φ))))
17 ax-sscoll 9417 . . . 4 uv𝑐z(x u y v φ𝑑 𝑐 y(y 𝑑x u φ))
1817spi 1426 . . 3 v𝑐z(x u y v φ𝑑 𝑐 y(y 𝑑x u φ))
1918spi 1426 . 2 𝑐z(x u y v φ𝑑 𝑐 y(y 𝑑x u φ))
2016, 19ch2varv 9223 1 𝑐z(x 𝑎 y 𝑏 φ𝑑 𝑐 y(y 𝑑x 𝑎 φ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  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-sscoll 9417
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: (None)
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