Theorem sbcom2 1980

Description: Commutativity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 27-May-1997.) (Proof modified to be intuitionistic by Jim Kingdon, 19-Feb-2018.)

Assertion

Ref	Expression
sbcom2	⊢ ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)

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

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

Proof of Theorem sbcom2

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcom2v2 1979	. . . 4 ⊢ ([𝑣 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑣 / 𝑧]𝜑)
2	1	sbbii 1758	. . 3 ⊢ ([𝑤 / 𝑣][𝑣 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑤 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑧]𝜑)
3		sbcom2v2 1979	. . 3 ⊢ ([𝑤 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑣][𝑣 / 𝑧]𝜑)
4	2, 3	bitri 183	. 2 ⊢ ([𝑤 / 𝑣][𝑣 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑣][𝑣 / 𝑧]𝜑)
5		ax-17 1519	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑣[𝑦 / 𝑥]𝜑)
6	5	sbco2vh 1938	. 2 ⊢ ([𝑤 / 𝑣][𝑣 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑)
7		ax-17 1519	. . . 4 ⊢ (𝜑 → ∀𝑣𝜑)
8	7	sbco2vh 1938	. . 3 ⊢ ([𝑤 / 𝑣][𝑣 / 𝑧]𝜑 ↔ [𝑤 / 𝑧]𝜑)
9	8	sbbii 1758	. 2 ⊢ ([𝑦 / 𝑥][𝑤 / 𝑣][𝑣 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)
10	4, 6, 9	3bitr3i 209	1 ⊢ ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)

Syntax hints:   ↔ wb 104  [wsb 1755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756

This theorem is referenced by:  2sb5rf  1982  2sb6rf  1983  sbco4lem  1999  sbco4  2000  sbmo  2078  cnvopab  5012