Theorem sbcom4 2126

Description: Commutativity law for substitution. This theorem was incorrectly used as our previous version of pm11.07 2127 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   𝜑,𝑧

Allowed substitution hints:   𝜑(𝑦, 𝑤)

Proof of Theorem sbcom4

Step	Hyp	Ref	Expression
1		sbv 2125	. 2 ⊢ ([𝑤 / 𝑥]𝜑 ↔ 𝜑)
2		sbv 2125	. . 3 ⊢ ([𝑦 / 𝑧]𝜑 ↔ 𝜑)
3	2	sbbii 2113	. 2 ⊢ ([𝑤 / 𝑥][𝑦 / 𝑧]𝜑 ↔ [𝑤 / 𝑥]𝜑)
4		sbv 2125	. . . 4 ⊢ ([𝑤 / 𝑧]𝜑 ↔ 𝜑)
5	4	sbbii 2113	. . 3 ⊢ ([𝑦 / 𝑥][𝑤 / 𝑧]𝜑 ↔ [𝑦 / 𝑥]𝜑)
6		sbv 2125	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)
7	5, 6	bitri 278	. 2 ⊢ ([𝑦 / 𝑥][𝑤 / 𝑧]𝜑 ↔ 𝜑)
8	1, 3, 7	3bitr4i 306	1 ⊢ ([𝑤 / 𝑥][𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)

This proof depends on syntax axioms:   ↔ wb 209  [wsb 2099

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100

This theorem is used by: (None)