Theorem sbcomv 2000

Description: Version of sbcom 2004 with a distinct variable constraint between 𝑥 and 𝑧. (Contributed by Jim Kingdon, 28-Feb-2018.)

Assertion

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

Distinct variable group:   𝑥,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem sbcomv

Step	Hyp	Ref	Expression
1		sbco3v 1998	. 2 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑥 / 𝑧]𝜑)
2		sbcocom 1999	. 2 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑)
3		sbcocom 1999	. 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑)
4	1, 2, 3	3bitr3i 210	1 ⊢ ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]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-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: (None)