Theorem sbco3 2088

Description: A composition law for substitution. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
sbco3	⊢ ([z / y][y / x]φ ↔ [z / x][x / y]φ)

Proof of Theorem sbco3

Step	Hyp	Ref	Expression
1		drsb1 2022	. . 3 ⊢ (∀x x = y → ([z / x][y / x]φ ↔ [z / y][y / x]φ))
2		sbequ12a 1921	. . . . 5 ⊢ (x = y → ([y / x]φ ↔ [x / y]φ))
3	2	alimi 1559	. . . 4 ⊢ (∀x x = y → ∀x([y / x]φ ↔ [x / y]φ))
4		spsbbi 2077	. . . 4 ⊢ (∀x([y / x]φ ↔ [x / y]φ) → ([z / x][y / x]φ ↔ [z / x][x / y]φ))
5	3, 4	syl 15	. . 3 ⊢ (∀x x = y → ([z / x][y / x]φ ↔ [z / x][x / y]φ))
6	1, 5	bitr3d 246	. 2 ⊢ (∀x x = y → ([z / y][y / x]φ ↔ [z / x][x / y]φ))
7		sbco 2083	. . . 4 ⊢ ([x / y][y / x]φ ↔ [x / y]φ)
8	7	sbbii 1653	. . 3 ⊢ ([z / x][x / y][y / x]φ ↔ [z / x][x / y]φ)
9		nfnae 1956	. . . 4 ⊢ Ⅎy ¬ ∀x x = y
10		nfnae 1956	. . . 4 ⊢ Ⅎx ¬ ∀x x = y
11		nfsb2 2058	. . . 4 ⊢ (¬ ∀x x = y → Ⅎx[y / x]φ)
12	9, 10, 11	sbco2d 2087	. . 3 ⊢ (¬ ∀x x = y → ([z / x][x / y][y / x]φ ↔ [z / y][y / x]φ))
13	8, 12	syl5rbbr 251	. 2 ⊢ (¬ ∀x x = y → ([z / y][y / x]φ ↔ [z / x][x / y]φ))
14	6, 13	pm2.61i 156	1 ⊢ ([z / y][y / x]φ ↔ [z / x][x / y]φ)

Syntax hints:  ¬ wn 3   ↔ wb 176  ∀wal 1540  [wsb 1648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is referenced by: (None)