Theorem sbal1yz 1859

Description: Lemma for proving sbal1 1860. Same as sbal1 1860 but with an additional distinct variable constraint on y and z. (Contributed by Jim Kingdon, 23-Feb-2018.)

Assertion

Ref	Expression
sbal1yz	⊢ (¬ ∀x x = z → ([z / y]∀xφ ↔ ∀x[z / y]φ))

Distinct variable groups:   x,y   y,z

Allowed substitution hints:   φ(x,y,z)

Proof of Theorem sbal1yz

Step	Hyp	Ref	Expression
1		dveeq2or 1679	. . . . . 6 ⊢ (∀x x = z ∨ Ⅎx y = z)
2		equcom 1575	. . . . . . . . 9 ⊢ (y = z ↔ z = y)
3	2	nfbii 1342	. . . . . . . 8 ⊢ (Ⅎx y = z ↔ Ⅎx z = y)
4		19.21t 1456	. . . . . . . 8 ⊢ (Ⅎx z = y → (∀x(z = y → φ) ↔ (z = y → ∀xφ)))
5	3, 4	sylbi 114	. . . . . . 7 ⊢ (Ⅎx y = z → (∀x(z = y → φ) ↔ (z = y → ∀xφ)))
6	5	orim2i 665	. . . . . 6 ⊢ ((∀x x = z ∨ Ⅎx y = z) → (∀x x = z ∨ (∀x(z = y → φ) ↔ (z = y → ∀xφ))))
7	1, 6	ax-mp 7	. . . . 5 ⊢ (∀x x = z ∨ (∀x(z = y → φ) ↔ (z = y → ∀xφ)))
8	7	ori 629	. . . 4 ⊢ (¬ ∀x x = z → (∀x(z = y → φ) ↔ (z = y → ∀xφ)))
9	8	albidv 1687	. . 3 ⊢ (¬ ∀x x = z → (∀y∀x(z = y → φ) ↔ ∀y(z = y → ∀xφ)))
10		alcom 1347	. . . 4 ⊢ (∀y∀x(z = y → φ) ↔ ∀x∀y(z = y → φ))
11		sb6 1748	. . . . . 6 ⊢ ([z / y]φ ↔ ∀y(y = z → φ))
12	2	imbi1i 227	. . . . . . 7 ⊢ ((y = z → φ) ↔ (z = y → φ))
13	12	albii 1339	. . . . . 6 ⊢ (∀y(y = z → φ) ↔ ∀y(z = y → φ))
14	11, 13	bitri 173	. . . . 5 ⊢ ([z / y]φ ↔ ∀y(z = y → φ))
15	14	albii 1339	. . . 4 ⊢ (∀x[z / y]φ ↔ ∀x∀y(z = y → φ))
16	10, 15	bitr4i 176	. . 3 ⊢ (∀y∀x(z = y → φ) ↔ ∀x[z / y]φ)
17		sb6 1748	. . . 4 ⊢ ([z / y]∀xφ ↔ ∀y(y = z → ∀xφ))
18	2	imbi1i 227	. . . . 5 ⊢ ((y = z → ∀xφ) ↔ (z = y → ∀xφ))
19	18	albii 1339	. . . 4 ⊢ (∀y(y = z → ∀xφ) ↔ ∀y(z = y → ∀xφ))
20	17, 19	bitr2i 174	. . 3 ⊢ (∀y(z = y → ∀xφ) ↔ [z / y]∀xφ)
21	9, 16, 20	3bitr3g 211	. 2 ⊢ (¬ ∀x x = z → (∀x[z / y]φ ↔ [z / y]∀xφ))
22	21	bicomd 129	1 ⊢ (¬ ∀x x = z → ([z / y]∀xφ ↔ ∀x[z / y]φ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98   ∨ wo 616  ∀wal 1226  Ⅎwnf 1329  [wsb 1627

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-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410

This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628

This theorem is referenced by:  sbal1  1860