Theorem sbcied2 3084

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

Hypotheses

Ref	Expression
sbcied2.1	⊢ (φ → A ∈ V)
sbcied2.2	⊢ (φ → A = B)
sbcied2.3	⊢ ((φ ∧ x = B) → (ψ ↔ χ))

Assertion

Ref	Expression
sbcied2	⊢ (φ → ([̣A / x]̣ψ ↔ χ))

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

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

Proof of Theorem sbcied2

Step	Hyp	Ref	Expression
1		sbcied2.1	. 2 ⊢ (φ → A ∈ V)
2		id 19	. . . 4 ⊢ (x = A → x = A)
3		sbcied2.2	. . . 4 ⊢ (φ → A = B)
4	2, 3	sylan9eqr 2407	. . 3 ⊢ ((φ ∧ x = A) → x = B)
5		sbcied2.3	. . 3 ⊢ ((φ ∧ x = B) → (ψ ↔ χ))
6	4, 5	syldan 456	. 2 ⊢ ((φ ∧ x = A) → (ψ ↔ χ))
7	1, 6	sbcied 3083	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: (None)