Theorem sbco 2083

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

Assertion

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

Proof of Theorem sbco

Step	Hyp	Ref	Expression
1		equsb2 2035	. . 3 ⊢ [y / x]y = x
2		sbequ12 1919	. . . . 5 ⊢ (y = x → (φ ↔ [x / y]φ))
3	2	bicomd 192	. . . 4 ⊢ (y = x → ([x / y]φ ↔ φ))
4	3	sbimi 1652	. . 3 ⊢ ([y / x]y = x → [y / x]([x / y]φ ↔ φ))
5	1, 4	ax-mp 5	. 2 ⊢ [y / x]([x / y]φ ↔ φ)
6		sbbi 2071	. 2 ⊢ ([y / x]([x / y]φ ↔ φ) ↔ ([y / x][x / y]φ ↔ [y / x]φ))
7	5, 6	mpbi 199	1 ⊢ ([y / x][x / y]φ ↔ [y / x]φ)

Syntax hints:   ↔ wb 176  [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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is referenced by:  sbid2  2084  sbco3  2088  sb9i  2094