Theorem csbeq2d 3160

Description: Formula-building deduction rule for class substitution. (Contributed by NM, 22-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)

Hypotheses

Ref	Expression
csbeq2d.1	⊢ Ⅎxφ
csbeq2d.2	⊢ (φ → B = C)

Assertion

Ref	Expression
csbeq2d	⊢ (φ → [A / x]B = [A / x]C)

Proof of Theorem csbeq2d

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq2d.1	. . . 4 ⊢ Ⅎxφ
2		csbeq2d.2	. . . . 5 ⊢ (φ → B = C)
3	2	eleq2d 2420	. . . 4 ⊢ (φ → (y ∈ B ↔ y ∈ C))
4	1, 3	sbcbid 3099	. . 3 ⊢ (φ → ([̣A / x]̣y ∈ B ↔ [̣A / x]̣y ∈ C))
5	4	abbidv 2467	. 2 ⊢ (φ → {y ∣ [̣A / x]̣y ∈ B} = {y ∣ [̣A / x]̣y ∈ C})
6		df-csb 3137	. 2 ⊢ [A / x]B = {y ∣ [̣A / x]̣y ∈ B}
7		df-csb 3137	. 2 ⊢ [A / x]C = {y ∣ [̣A / x]̣y ∈ C}
8	5, 6, 7	3eqtr4g 2410	1 ⊢ (φ → [A / x]B = [A / x]C)

Syntax hints:   → wi 4  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  {cab 2339  [̣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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-sbc 3047  df-csb 3137

This theorem is referenced by:  csbeq2dv  3161