Theorem sbco2yz 2023

Description: This is a version of sbco2 2025 where 𝑧 is distinct from 𝑦. It is a lemma on the way to proving sbco2 2025 which has no distinct variable constraints. (Contributed by Jim Kingdon, 19-Mar-2018.)

Hypothesis

Ref	Expression
sbco2yz.1	⊢ Ⅎ𝑧𝜑

Assertion

Ref	Expression
sbco2yz	⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Distinct variable group:   𝑦,𝑧

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

Proof of Theorem sbco2yz

Step	Hyp	Ref	Expression
1		sbco2yz.1	. . . 4 ⊢ Ⅎ𝑧𝜑
2	1	nfsb 2006	. . 3 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑
3	2	nfri 1572	. 2 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)
4		sbequ 1893	. 2 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
5	3, 4	sbieh 1843	1 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 105  Ⅎwnf 1513  [wsb 1815

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816

This theorem is referenced by:  sbco2h  2024