Theorem sbco3 1256

Description: A composition law for substitution.

Assertion

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

Proof of Theorem sbco3

Step	Hyp	Ref	Expression
1		drsb1 1174	. . 3 ⊢ (∀x x = y → ([z / x][y / x]φ ↔ [z / y][y / x]φ))
2		sbequ12a 1182	. . . . 5 ⊢ (x = y → ([y / x]φ ↔ [x / y]φ))
3	2	19.20i 991	. . . 4 ⊢ (∀x x = y → ∀x([y / x]φ ↔ [x / y]φ))
4		a4sbbi 1244	. . . 4 ⊢ (∀x([y / x]φ ↔ [x / y]φ) → ([z / x][y / x]φ ↔ [z / x][x / y]φ))
5	3, 4	syl 10	. . 3 ⊢ (∀x x = y → ([z / x][y / x]φ ↔ [z / x][x / y]φ))
6	1, 5	bitr3d 529	. 2 ⊢ (∀x x = y → ([z / y][y / x]φ ↔ [z / x][x / y]φ))
7		hbnae 1146	. . . 4 ⊢ (¬ ∀x x = y → ∀y ¬ ∀x x = y)
8		hbnae 1146	. . . 4 ⊢ (¬ ∀x x = y → ∀x ¬ ∀x x = y)
9		hbsb2 1226	. . . 4 ⊢ (¬ ∀x x = y → ([y / x]φ → ∀x[y / x]φ))
10	7, 8, 9	sbco2d 1255	. . 3 ⊢ (¬ ∀x x = y → ([z / x][x / y][y / x]φ ↔ [z / y][y / x]φ))
11		sbco 1251	. . . 4 ⊢ ([x / y][y / x]φ ↔ [x / y]φ)
12	11	sbbii 1173	. . 3 ⊢ ([z / x][x / y][y / x]φ ↔ [z / x][x / y]φ)
13	10, 12	syl5rbbr 534	. 2 ⊢ (¬ ∀x x = y → ([z / y][y / x]φ ↔ [z / x][x / y]φ))
14	6, 13	pm2.61i 126	1 ⊢ ([z / y][y / x]φ ↔ [z / x][x / y]φ)

Syntax hints:  ¬ wn 2   ↔ wb 146  ∀wal 953  [wsbc 1169

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-11o 1217

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171

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