Theorem sbco3 2003

Description: A composition law for substitution. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 22-Mar-2018.)

Assertion

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

Proof of Theorem sbco3

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbco3xzyz 2002	. . 3 ⊢ ([𝑤 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑤 / 𝑥][𝑥 / 𝑦]𝜑)
2	1	sbbii 1789	. 2 ⊢ ([𝑧 / 𝑤][𝑤 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑤][𝑤 / 𝑥][𝑥 / 𝑦]𝜑)
3		ax-17 1550	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑤[𝑦 / 𝑥]𝜑)
4	3	sbco2h 1993	. 2 ⊢ ([𝑧 / 𝑤][𝑤 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑)
5		ax-17 1550	. . 3 ⊢ ([𝑥 / 𝑦]𝜑 → ∀𝑤[𝑥 / 𝑦]𝜑)
6	5	sbco2h 1993	. 2 ⊢ ([𝑧 / 𝑤][𝑤 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑)
7	2, 4, 6	3bitr3i 210	1 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑)

Syntax hints:   ↔ wb 105  [wsb 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787

This theorem is referenced by:  sbcom  2004