Theorem sbcom2v2 2015

Description: Lemma for proving sbcom2 2016. It is the same as sbcom2v 2014 but removes the distinct variable constraint on 𝑥 and 𝑦. (Contributed by Jim Kingdon, 19-Feb-2018.)

Assertion

Ref	Expression
sbcom2v2	⊢ ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)

Distinct variable groups:   𝑥,𝑤,𝑧   𝑦,𝑧

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

Proof of Theorem sbcom2v2

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcom2v 2014	. . 3 ⊢ ([𝑤 / 𝑧][𝑦 / 𝑣][𝑣 / 𝑥]𝜑 ↔ [𝑦 / 𝑣][𝑤 / 𝑧][𝑣 / 𝑥]𝜑)
2		sbcom2v 2014	. . . 4 ⊢ ([𝑤 / 𝑧][𝑣 / 𝑥]𝜑 ↔ [𝑣 / 𝑥][𝑤 / 𝑧]𝜑)
3	2	sbbii 1789	. . 3 ⊢ ([𝑦 / 𝑣][𝑤 / 𝑧][𝑣 / 𝑥]𝜑 ↔ [𝑦 / 𝑣][𝑣 / 𝑥][𝑤 / 𝑧]𝜑)
4	1, 3	bitri 184	. 2 ⊢ ([𝑤 / 𝑧][𝑦 / 𝑣][𝑣 / 𝑥]𝜑 ↔ [𝑦 / 𝑣][𝑣 / 𝑥][𝑤 / 𝑧]𝜑)
5		ax-17 1550	. . . 4 ⊢ (𝜑 → ∀𝑣𝜑)
6	5	sbco2vh 1974	. . 3 ⊢ ([𝑦 / 𝑣][𝑣 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
7	6	sbbii 1789	. 2 ⊢ ([𝑤 / 𝑧][𝑦 / 𝑣][𝑣 / 𝑥]𝜑 ↔ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑)
8		ax-17 1550	. . 3 ⊢ ([𝑤 / 𝑧]𝜑 → ∀𝑣[𝑤 / 𝑧]𝜑)
9	8	sbco2vh 1974	. 2 ⊢ ([𝑦 / 𝑣][𝑣 / 𝑥][𝑤 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)
10	4, 7, 9	3bitr3i 210	1 ⊢ ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 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:  sbcom2  2016