Theorem sbcco 3069

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 3056	. 2 ⊢ ([̣A / y]̣[̣y / x]̣φ → A ∈ V)
2		sbcex 3056	. 2 ⊢ ([̣A / x]̣φ → A ∈ V)
3		dfsbcq 3049	. . 3 ⊢ (z = A → ([̣z / y]̣[̣y / x]̣φ ↔ [̣A / y]̣[̣y / x]̣φ))
4		dfsbcq 3049	. . 3 ⊢ (z = A → ([̣z / x]̣φ ↔ [̣A / x]̣φ))
5		sbsbc 3051	. . . . . 6 ⊢ ([y / x]φ ↔ [̣y / x]̣φ)
6	5	sbbii 1653	. . . . 5 ⊢ ([z / y][y / x]φ ↔ [z / y][̣y / x]̣φ)
7		nfv 1619	. . . . . 6 ⊢ Ⅎyφ
8	7	sbco2 2086	. . . . 5 ⊢ ([z / y][y / x]φ ↔ [z / x]φ)
9		sbsbc 3051	. . . . 5 ⊢ ([z / y][̣y / x]̣φ ↔ [̣z / y]̣[̣y / x]̣φ)
10	6, 8, 9	3bitr3ri 267	. . . 4 ⊢ ([̣z / y]̣[̣y / x]̣φ ↔ [z / x]φ)
11		sbsbc 3051	. . . 4 ⊢ ([z / x]φ ↔ [̣z / x]̣φ)
12	10, 11	bitri 240	. . 3 ⊢ ([̣z / y]̣[̣y / x]̣φ ↔ [̣z / x]̣φ)
13	3, 4, 12	vtoclbg 2916	. 2 ⊢ (A ∈ V → ([̣A / y]̣[̣y / x]̣φ ↔ [̣A / x]̣φ))
14	1, 2, 13	pm5.21nii 342	1 ⊢ ([̣A / y]̣[̣y / x]̣φ ↔ [̣A / x]̣φ)

This proof depends on syntax axioms:   ↔ wb 176  [wsb 1648   ∈ wcel 1710  Vcvv 2860  [̣wsbc 3047

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048

This theorem is used by:  sbc7  3074  sbccom  3118  sbcralt  3119  csbco  3146