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Theorem sscoll2 7402
Description: Version of ax-sscoll 7401 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 1398 . . 3 𝑐(u = 𝑎 v = 𝑏)
2 nfv 1398 . . . 4 z(u = 𝑎 v = 𝑏)
3 simpl 102 . . . . . 6 ((u = 𝑎 v = 𝑏) → u = 𝑎)
4 rexeq 2480 . . . . . . 7 (v = 𝑏 → (y v φy 𝑏 φ))
54adantl 262 . . . . . 6 ((u = 𝑎 v = 𝑏) → (y v φy 𝑏 φ))
63, 5raleqbidv 2491 . . . . 5 ((u = 𝑎 v = 𝑏) → (x u y v φx 𝑎 y 𝑏 φ))
7 nfv 1398 . . . . . 6 𝑑(u = 𝑎 v = 𝑏)
8 nfv 1398 . . . . . . 7 y(u = 𝑎 v = 𝑏)
9 rexeq 2480 . . . . . . . . 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 1484 . . . . . 6 ((u = 𝑎 v = 𝑏) → (y(y 𝑑x u φ) ↔ y(y 𝑑x 𝑎 φ)))
137, 12rexbid 2299 . . . . 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 1484 . . 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 1485 . 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 7401 . . . 4 uv𝑐z(x u y v φ𝑑 𝑐 y(y 𝑑x u φ))
1817spi 1407 . . 3 v𝑐z(x u y v φ𝑑 𝑐 y(y 𝑑x u φ))
1918spi 1407 . 2 𝑐z(x u y v φ𝑑 𝑐 y(y 𝑑x u φ))
2016, 19ch2varv 7207 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 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-sscoll 7401
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: (None)
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