Theorem sbciedf 3082

Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 29-Dec-2014.)

Hypotheses

Ref	Expression
sbcied.1	⊢ (φ → A ∈ V)
sbcied.2	⊢ ((φ ∧ x = A) → (ψ ↔ χ))
sbciedf.3	⊢ Ⅎxφ
sbciedf.4	⊢ (φ → Ⅎxχ)

Assertion

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

Distinct variable group:   x,A

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

Proof of Theorem sbciedf

Step	Hyp	Ref	Expression
1		sbcied.1	. 2 ⊢ (φ → A ∈ V)
2		sbciedf.4	. 2 ⊢ (φ → Ⅎxχ)
3		sbciedf.3	. . 3 ⊢ Ⅎxφ
4		sbcied.2	. . . 4 ⊢ ((φ ∧ x = A) → (ψ ↔ χ))
5	4	ex 423	. . 3 ⊢ (φ → (x = A → (ψ ↔ χ)))
6	3, 5	alrimi 1765	. 2 ⊢ (φ → ∀x(x = A → (ψ ↔ χ)))
7		sbciegft 3077	. 2 ⊢ ((A ∈ V ∧ Ⅎxχ ∧ ∀x(x = A → (ψ ↔ χ))) → ([̣A / x]̣ψ ↔ χ))
8	1, 2, 6, 7	syl3anc 1182	1 ⊢ (φ → ([̣A / x]̣ψ ↔ χ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  [̣wsbc 3047

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-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 used by:  sbcied  3083  sbc2iegf  3113  csbiebt  3173  sbcnestgf  3184