Theorem sbco2h 1893

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 1403	. . . 4 ⊢ Ⅎ𝑧𝜑
3	2	sbco2yz 1892	. . 3 ⊢ ([𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝜑)
4	3	sbbii 1702	. 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑤][𝑤 / 𝑥]𝜑)
5		nfv 1473	. . 3 ⊢ Ⅎ𝑤[𝑧 / 𝑥]𝜑
6	5	sbco2yz 1892	. 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧][𝑧 / 𝑥]𝜑)
7		nfv 1473	. . 3 ⊢ Ⅎ𝑤𝜑
8	7	sbco2yz 1892	. 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
9	4, 6, 8	3bitr3i 209	1 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1294  [wsb 1699

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480

This theorem depends on definitions:  df-bi 116  df-nf 1402  df-sb 1700

This theorem is referenced by:  sbco2  1894  sbco2d  1895  sbco3  1903  elsb3  1907  elsb4  1908  sb9  1910