Theorem sbcco 3425

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 3412	. 2 ⊢ ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑 → 𝐴 ∈ V)
2		sbcex 3412	. 2 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)
3		dfsbcq 3404	. . 3 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑦][𝑦 / 𝑥]𝜑))
4		dfsbcq 3404	. . 3 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
5		sbsbc 3406	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
6	5	sbbii 1874	. . . . 5 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑)
7		nfv 1830	. . . . . 6 ⊢ Ⅎ𝑦𝜑
8	7	sbco2 2403	. . . . 5 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
9		sbsbc 3406	. . . . 5 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑)
10	6, 8, 9	3bitr3ri 290	. . . 4 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
11		sbsbc 3406	. . . 4 ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
12	10, 11	bitri 263	. . 3 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
13	3, 4, 12	vtoclbg 3240	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
14	1, 2, 13	pm5.21nii 367	1 ⊢ ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)

Syntax hints:   ↔ wb 195  [wsb 1867   ∈ wcel 1977  Vcvv 3173  [wsbc 3402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175  df-sbc 3403

This theorem is referenced by:  sbc7  3430  sbccom  3476  sbcralt  3477  csbco  3509  bnj62  30040  bnj610  30071  bnj976  30102  bnj1468  30170  sbccom2  33100  sbccom2f  33101  aomclem6  36647