Theorem sbco2 1965

Description: A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.)

Hypothesis

Ref	Expression
sbco2.1	⊢ Ⅎ𝑧𝜑

Assertion

Ref	Expression
sbco2	⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Proof of Theorem sbco2

Step	Hyp	Ref	Expression
1		sbco2.1	. . 3 ⊢ Ⅎ𝑧𝜑
2	1	nfri 1519	. 2 ⊢ (𝜑 → ∀𝑧𝜑)
3	2	sbco2h 1964	1 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 105  Ⅎwnf 1460  [wsb 1762

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763

This theorem is referenced by:  nfsbt  1976  sb7af  1993  sbco4lem  2006  sbco4  2007  eqsb1  2281  clelsb1  2282  clelsb2  2283  sb8ab  2299  clelsb1f  2323  sbralie  2723  sbcco  2986