Theorem sbal1 1875

Description: A theorem used in elimination of disjoint variable restriction on x and y by replacing it with a distinctor ¬ ∀xx = z. (Contributed by NM, 5-Aug-1993.) (Proof rewitten by Jim Kingdon, 24-Feb-2018.)

Assertion

Ref	Expression
sbal1	⊢ (¬ ∀x x = z → ([z / y]∀xφ ↔ ∀x[z / y]φ))

Distinct variable group:   x,y

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

Proof of Theorem sbal1

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbal 1873	. . . 4 ⊢ ([w / y]∀xφ ↔ ∀x[w / y]φ)
2	1	sbbii 1645	. . 3 ⊢ ([z / w][w / y]∀xφ ↔ [z / w]∀x[w / y]φ)
3		sbal1yz 1874	. . 3 ⊢ (¬ ∀x x = z → ([z / w]∀x[w / y]φ ↔ ∀x[z / w][w / y]φ))
4	2, 3	syl5bb 181	. 2 ⊢ (¬ ∀x x = z → ([z / w][w / y]∀xφ ↔ ∀x[z / w][w / y]φ))
5		ax-17 1416	. . 3 ⊢ (∀xφ → ∀w∀xφ)
6	5	sbco2v 1818	. 2 ⊢ ([z / w][w / y]∀xφ ↔ [z / y]∀xφ)
7		ax-17 1416	. . . 4 ⊢ (φ → ∀wφ)
8	7	sbco2v 1818	. . 3 ⊢ ([z / w][w / y]φ ↔ [z / y]φ)
9	8	albii 1356	. 2 ⊢ (∀x[z / w][w / y]φ ↔ ∀x[z / y]φ)
10	4, 6, 9	3bitr3g 211	1 ⊢ (¬ ∀x x = z → ([z / y]∀xφ ↔ ∀x[z / y]φ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98  ∀wal 1240  [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-bnd 1396  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: (None)