Theorem sbco2vh 2005

Description: This is a version of sbco2 2025 where 𝑧 is distinct from 𝑥. (Contributed by Jim Kingdon, 12-Feb-2018.)

Hypothesis

Ref	Expression
sbco2vh.1	⊢ (𝜑 → ∀𝑧𝜑)

Assertion

Ref	Expression
sbco2vh	⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Distinct variable group:   𝑥,𝑧

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

Proof of Theorem sbco2vh

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbco2vh.1	. . . 4 ⊢ (𝜑 → ∀𝑧𝜑)
2	1	sbco2vlem 2004	. . 3 ⊢ ([𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝜑)
3	2	sbbii 1818	. 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑤][𝑤 / 𝑥]𝜑)
4		ax-17 1579	. . 3 ⊢ ([𝑧 / 𝑥]𝜑 → ∀𝑤[𝑧 / 𝑥]𝜑)
5	4	sbco2vlem 2004	. 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧][𝑧 / 𝑥]𝜑)
6		ax-17 1579	. . 3 ⊢ (𝜑 → ∀𝑤𝜑)
7	6	sbco2vlem 2004	. 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
8	3, 5, 7	3bitr3i 210	1 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1400  [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-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:  nfsb  2006  equsb3  2011  sbn  2012  sbim  2013  sbor  2014  sban  2015  sbco2vd  2027  sbco3v  2029  sbcom2v2  2046  sbcom2  2047  dfsb7  2051  sb7f  2052  sbal  2060  sbal1  2062  sbex  2064