Theorem sbco2h 1976

Description: A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof rewritten by Jim Kingdon, 19-Mar-2018.)

Hypothesis

Ref	Expression
sbco2h.1	⊢ (𝜑 → ∀𝑧𝜑)

Assertion

Ref	Expression
sbco2h	⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Proof of Theorem sbco2h

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbco2h.1	. . . . 5 ⊢ (𝜑 → ∀𝑧𝜑)
2	1	nfi 1473	. . . 4 ⊢ Ⅎ𝑧𝜑
3	2	sbco2yz 1975	. . 3 ⊢ ([𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝜑)
4	3	sbbii 1776	. 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑤][𝑤 / 𝑥]𝜑)
5		nfv 1539	. . 3 ⊢ Ⅎ𝑤[𝑧 / 𝑥]𝜑
6	5	sbco2yz 1975	. 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧][𝑧 / 𝑥]𝜑)
7		nfv 1539	. . 3 ⊢ Ⅎ𝑤𝜑
8	7	sbco2yz 1975	. 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
9	4, 6, 8	3bitr3i 210	1 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1362  [wsb 1773

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774

This theorem is referenced by:  sbco2  1977  sbco2d  1978  sbco3  1986  sb9  1991  elsb1  2167  elsb2  2168