Theorem sbcied 3083

Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.)

Hypotheses

Ref	Expression
sbcied.1	⊢ (φ → A ∈ V)
sbcied.2	⊢ ((φ ∧ x = A) → (ψ ↔ χ))

Assertion

Ref	Expression
sbcied	⊢ (φ → ([̣A / x]̣ψ ↔ χ))

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

Allowed substitution hints:   ψ(x)   V(x)

Proof of Theorem sbcied

Step	Hyp	Ref	Expression
1		sbcied.1	. 2 ⊢ (φ → A ∈ V)
2		sbcied.2	. 2 ⊢ ((φ ∧ x = A) → (ψ ↔ χ))
3		nfv 1619	. 2 ⊢ Ⅎxφ
4		nfvd 1620	. 2 ⊢ (φ → Ⅎxχ)
5	1, 2, 3, 4	sbciedf 3082	1 ⊢ (φ → ([̣A / x]̣ψ ↔ χ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ 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-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 2479  df-v 2862  df-sbc 3048

This theorem is referenced by:  sbcied2  3084  sbc2iedv  3115  sbc3ie  3116  sbcralt  3119