Theorem sbcom4 2446

Description: Commutativity law for substitution. This theorem was incorrectly used as our previous version of pm11.07 2447 but may still be useful. (Contributed by Andrew Salmon, 17-Jun-2011.) (Proof shortened by Jim Kingdon, 22-Jan-2018.)

Assertion

Ref	Expression
sbcom4	⊢ ([𝑤 / 𝑥][𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)

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

Allowed substitution hint:   𝜑(𝑤)

Proof of Theorem sbcom4

Step	Hyp	Ref	Expression
1		nfv 1843	. . 3 ⊢ Ⅎ𝑥𝜑
2	1	sbf 2380	. 2 ⊢ ([𝑤 / 𝑥]𝜑 ↔ 𝜑)
3		nfv 1843	. . . 4 ⊢ Ⅎ𝑧𝜑
4	3	sbf 2380	. . 3 ⊢ ([𝑦 / 𝑧]𝜑 ↔ 𝜑)
5	4	sbbii 1887	. 2 ⊢ ([𝑤 / 𝑥][𝑦 / 𝑧]𝜑 ↔ [𝑤 / 𝑥]𝜑)
6	3	sbf 2380	. . . 4 ⊢ ([𝑤 / 𝑧]𝜑 ↔ 𝜑)
7	6	sbbii 1887	. . 3 ⊢ ([𝑦 / 𝑥][𝑤 / 𝑧]𝜑 ↔ [𝑦 / 𝑥]𝜑)
8	1	sbf 2380	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)
9	7, 8	bitri 264	. 2 ⊢ ([𝑦 / 𝑥][𝑤 / 𝑧]𝜑 ↔ 𝜑)
10	2, 5, 9	3bitr4i 292	1 ⊢ ([𝑤 / 𝑥][𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)

Syntax hints:   ↔ wb 196  [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by: (None)