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Theorem strcollnfALT 9446
Description: Alternate proof of strcollnf 9445, not using strcollnft 9444. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
strcollnf.nf 𝑏φ
Assertion
Ref Expression
strcollnfALT (x 𝑎 yφ𝑏y(y 𝑏x 𝑎 φ))
Distinct variable group:   𝑎,𝑏,x,y
Allowed substitution hints:   φ(x,y,𝑎,𝑏)

Proof of Theorem strcollnfALT
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 strcoll2 9443 . 2 (x 𝑎 yφzy(y zx 𝑎 φ))
2 nfv 1418 . . . . 5 𝑏 y z
3 nfcv 2175 . . . . . 6 𝑏𝑎
4 strcollnf.nf . . . . . 6 𝑏φ
53, 4nfrexxy 2355 . . . . 5 𝑏x 𝑎 φ
62, 5nfbi 1478 . . . 4 𝑏(y zx 𝑎 φ)
76nfal 1465 . . 3 𝑏y(y zx 𝑎 φ)
8 nfv 1418 . . 3 zy(y 𝑏x 𝑎 φ)
9 elequ2 1598 . . . . 5 (z = 𝑏 → (y zy 𝑏))
109bibi1d 222 . . . 4 (z = 𝑏 → ((y zx 𝑎 φ) ↔ (y 𝑏x 𝑎 φ)))
1110albidv 1702 . . 3 (z = 𝑏 → (y(y zx 𝑎 φ) ↔ y(y 𝑏x 𝑎 φ)))
127, 8, 11cbvex 1636 . 2 (zy(y zx 𝑎 φ) ↔ 𝑏y(y 𝑏x 𝑎 φ))
131, 12sylib 127 1 (x 𝑎 yφ𝑏y(y 𝑏x 𝑎 φ))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1240  wnf 1346  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-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-strcoll 9442
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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