Theorem csbie2g 3183

Description: Conversion of implicit substitution to explicit class substitution. This version of sbcie 3081 avoids a disjointness condition on x, A by substituting twice. (Contributed by Mario Carneiro, 11-Nov-2016.)

Hypotheses

Ref	Expression
csbie2g.1	⊢ (x = y → B = C)
csbie2g.2	⊢ (y = A → C = D)

Assertion

Ref	Expression
csbie2g	⊢ (A ∈ V → [A / x]B = D)

Distinct variable groups:   x,y   y,A   y,B   x,C   y,D

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

Proof of Theorem csbie2g

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-csb 3138	. 2 ⊢ [A / x]B = {z ∣ [̣A / x]̣z ∈ B}
2		csbie2g.1	. . . . 5 ⊢ (x = y → B = C)
3	2	eleq2d 2420	. . . 4 ⊢ (x = y → (z ∈ B ↔ z ∈ C))
4		csbie2g.2	. . . . 5 ⊢ (y = A → C = D)
5	4	eleq2d 2420	. . . 4 ⊢ (y = A → (z ∈ C ↔ z ∈ D))
6	3, 5	sbcie2g 3080	. . 3 ⊢ (A ∈ V → ([̣A / x]̣z ∈ B ↔ z ∈ D))
7	6	eqabcdv 2470	. 2 ⊢ (A ∈ V → {z ∣ [̣A / x]̣z ∈ B} = D)
8	1, 7	syl5eq 2397	1 ⊢ (A ∈ V → [A / x]B = D)

This proof depends on syntax axioms:   → wi 4   = wceq 1642   ∈ wcel 1710  {cab 2339  [̣wsbc 3047  [csb 3137

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  df-csb 3138

This theorem is used by: (None)