Theorem sbal1yz 1874

Description: Lemma for proving sbal1 1875. Same as sbal1 1875 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 1694	. . . . . 6 ⊢ (∀x x = z ∨ Ⅎx y = z)
2		equcom 1590	. . . . . . . . 9 ⊢ (y = z ↔ z = y)
3	2	nfbii 1359	. . . . . . . 8 ⊢ (Ⅎx y = z ↔ Ⅎx z = y)
4		19.21t 1471	. . . . . . . 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 677	. . . . . 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 641	. . . 4 ⊢ (¬ ∀x x = z → (∀x(z = y → φ) ↔ (z = y → ∀xφ)))
9	8	albidv 1702	. . 3 ⊢ (¬ ∀x x = z → (∀y∀x(z = y → φ) ↔ ∀y(z = y → ∀xφ)))
10		alcom 1364	. . . 4 ⊢ (∀y∀x(z = y → φ) ↔ ∀x∀y(z = y → φ))
11		sb6 1763	. . . . . 6 ⊢ ([z / y]φ ↔ ∀y(y = z → φ))
12	2	imbi1i 227	. . . . . . 7 ⊢ ((y = z → φ) ↔ (z = y → φ))
13	12	albii 1356	. . . . . 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 1356	. . . 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 1763	. . . 4 ⊢ ([z / y]∀xφ ↔ ∀y(y = z → ∀xφ))
18	2	imbi1i 227	. . . . 5 ⊢ ((y = z → ∀xφ) ↔ (z = y → ∀xφ))
19	18	albii 1356	. . . 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 628  ∀wal 1240  Ⅎwnf 1346  [wsb 1642

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 545  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-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643

This theorem is referenced by:  sbal1  1875