Theorem sbco 910

Description: A composition law for substitution.

Assertion

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

Proof of Theorem sbco

Step	Hyp	Ref	Expression
1		sbeq2 901	. . 3 ⊢ [y / x]y = x
2		sbequ12 865	. . . . 5 ⊢ (y = x → (φ ↔ [x / y]φ))
3	2	bicomd 399	. . . 4 ⊢ (y = x → ([x / y]φ ↔ φ))
4	3	sbimi 854	. . 3 ⊢ ([y / x]y = x → [y / x]([x / y]φ ↔ φ))
5	1, 4	ax-mp 6	. 2 ⊢ [y / x]([x / y]φ ↔ φ)
6		sbbi 890	. 2 ⊢ ([y / x]([x / y]φ ↔ φ) ↔ ([y / x][x / y]φ ↔ [y / x]φ))
7	5, 6	mpbi 164	1 ⊢ ([y / x][x / y]φ ↔ [y / x]φ)

Syntax hints:   ↔ wb 127   = weq 797  [wsb 852

This theorem is referenced by:  sbid2 911  sbco3 915  sb5f1 917  sb9i 920

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853

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