Theorem sbccsbg 3164

Description: Substitution into a wff expressed in terms of substitution into a class. (Contributed by NM, 15-Aug-2007.)

Assertion

Ref	Expression
sbccsbg	⊢ (A ∈ V → ([̣A / x]̣φ ↔ y ∈ [A / x]{y ∣ φ}))

Distinct variable group:   x,y

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

Proof of Theorem sbccsbg

Step	Hyp	Ref	Expression
1		abid 2341	. . 3 ⊢ (y ∈ {y ∣ φ} ↔ φ)
2	1	sbcbii 3101	. 2 ⊢ ([̣A / x]̣y ∈ {y ∣ φ} ↔ [̣A / x]̣φ)
3		sbcel2g 3157	. 2 ⊢ (A ∈ V → ([̣A / x]̣y ∈ {y ∣ φ} ↔ y ∈ [A / x]{y ∣ φ}))
4	2, 3	syl5bbr 250	1 ⊢ (A ∈ V → ([̣A / x]̣φ ↔ y ∈ [A / x]{y ∣ φ}))

Syntax hints:   → wi 4   ↔ wb 176   ∈ 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-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 2478  df-v 2861  df-sbc 3047  df-csb 3137

This theorem is referenced by: (None)