Theorem sbcexg 3097

Description: Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.)

Assertion

Ref	Expression
sbcexg	⊢ (A ∈ V → ([̣A / y]̣∃xφ ↔ ∃x[̣A / y]̣φ))

Distinct variable groups:   x,A   x,y

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

Proof of Theorem sbcexg

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 3050	. 2 ⊢ (z = A → ([z / y]∃xφ ↔ [̣A / y]̣∃xφ))
2		dfsbcq2 3050	. . 3 ⊢ (z = A → ([z / y]φ ↔ [̣A / y]̣φ))
3	2	exbidv 1626	. 2 ⊢ (z = A → (∃x[z / y]φ ↔ ∃x[̣A / y]̣φ))
4		sbex 2128	. 2 ⊢ ([z / y]∃xφ ↔ ∃x[z / y]φ)
5	1, 3, 4	vtoclbg 2916	1 ⊢ (A ∈ V → ([̣A / y]̣∃xφ ↔ ∃x[̣A / y]̣φ))

Syntax hints:   → wi 4   ↔ wb 176  ∃wex 1541   = wceq 1642  [wsb 1648   ∈ wcel 1710  [̣wsbc 3047

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 2479  df-v 2862  df-sbc 3048

This theorem is referenced by:  sbcabel  3124  csbunig  3900  csbxpg  4814  csbrng  4967