Theorem sbcco3g 3191

Description: Composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)

Hypothesis

Ref	Expression
sbcco3g.1	⊢ (x = A → B = C)

Assertion

Ref	Expression
sbcco3g	⊢ (A ∈ V → ([̣A / x]̣[̣B / y]̣φ ↔ [̣C / y]̣φ))

Distinct variable groups:   x,A   φ,x   x,C

Allowed substitution hints:   φ(y)   A(y)   B(x,y)   C(y)   V(x,y)

Proof of Theorem sbcco3g

Step	Hyp	Ref	Expression
1		sbcnestg 3185	. 2 ⊢ (A ∈ V → ([̣A / x]̣[̣B / y]̣φ ↔ [̣[A / x]B / y]̣φ))
2		elex 2867	. . 3 ⊢ (A ∈ V → A ∈ V)
3		nfcvd 2490	. . . 4 ⊢ (A ∈ V → ℲxC)
4		sbcco3g.1	. . . 4 ⊢ (x = A → B = C)
5	3, 4	csbiegf 3176	. . 3 ⊢ (A ∈ V → [A / x]B = C)
6		dfsbcq 3048	. . 3 ⊢ ([A / x]B = C → ([̣[A / x]B / y]̣φ ↔ [̣C / y]̣φ))
7	2, 5, 6	3syl 18	. 2 ⊢ (A ∈ V → ([̣[A / x]B / y]̣φ ↔ [̣C / y]̣φ))
8	1, 7	bitrd 244	1 ⊢ (A ∈ V → ([̣A / x]̣[̣B / y]̣φ ↔ [̣C / y]̣φ))

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710  Vcvv 2859  [̣wsbc 3046  [csb 3136

This theorem was proved from 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 theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-sbc 3047  df-csb 3137

This theorem is referenced by:  sbcco3gOLD  3192