Theorem sbcom2v2 1986

Description: Lemma for proving sbcom2 1987. It is the same as sbcom2v 1985 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 1985	. . 3 ⊢ ([𝑤 / 𝑧][𝑦 / 𝑣][𝑣 / 𝑥]𝜑 ↔ [𝑦 / 𝑣][𝑤 / 𝑧][𝑣 / 𝑥]𝜑)
2		sbcom2v 1985	. . . 4 ⊢ ([𝑤 / 𝑧][𝑣 / 𝑥]𝜑 ↔ [𝑣 / 𝑥][𝑤 / 𝑧]𝜑)
3	2	sbbii 1765	. . 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 1526	. . . 4 ⊢ (𝜑 → ∀𝑣𝜑)
6	5	sbco2vh 1945	. . 3 ⊢ ([𝑦 / 𝑣][𝑣 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
7	6	sbbii 1765	. 2 ⊢ ([𝑤 / 𝑧][𝑦 / 𝑣][𝑣 / 𝑥]𝜑 ↔ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑)
8		ax-17 1526	. . 3 ⊢ ([𝑤 / 𝑧]𝜑 → ∀𝑣[𝑤 / 𝑧]𝜑)
9	8	sbco2vh 1945	. 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 1762

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763

This theorem is referenced by:  sbcom2  1987