Theorem sbcco 2976

Description: A composition law for class substitution. (Contributed by NM, 26-Sep-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)

Assertion

Ref	Expression
sbcco	⊢ ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)

Distinct variable group:   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥,𝑦)

Proof of Theorem sbcco

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcex 2963	. 2 ⊢ ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑 → 𝐴 ∈ V)
2		sbcex 2963	. 2 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)
3		dfsbcq 2957	. . 3 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑦][𝑦 / 𝑥]𝜑))
4		dfsbcq 2957	. . 3 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
5		sbsbc 2959	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
6	5	sbbii 1758	. . . . 5 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑)
7		nfv 1521	. . . . . 6 ⊢ Ⅎ𝑦𝜑
8	7	sbco2 1958	. . . . 5 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
9		sbsbc 2959	. . . . 5 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑)
10	6, 8, 9	3bitr3ri 210	. . . 4 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
11		sbsbc 2959	. . . 4 ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
12	10, 11	bitri 183	. . 3 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
13	3, 4, 12	vtoclbg 2791	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
14	1, 2, 13	pm5.21nii 699	1 ⊢ ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)

Syntax hints:   ↔ wb 104  [wsb 1755   ∈ wcel 2141  Vcvv 2730  [wsbc 2955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956

This theorem is referenced by:  sbc7  2981  sbccom  3030  sbcralt  3031  sbcrext  3032  csbco  3059