Theorem spcimdv 2937

Description: Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
spcimdv.1	⊢ (φ → A ∈ B)
spcimdv.2	⊢ ((φ ∧ x = A) → (ψ → χ))

Assertion

Ref	Expression
spcimdv	⊢ (φ → (∀xψ → χ))

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

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

Proof of Theorem spcimdv

Step	Hyp	Ref	Expression
1		spcimdv.2	. . . 4 ⊢ ((φ ∧ x = A) → (ψ → χ))
2	1	ex 423	. . 3 ⊢ (φ → (x = A → (ψ → χ)))
3	2	alrimiv 1631	. 2 ⊢ (φ → ∀x(x = A → (ψ → χ)))
4		spcimdv.1	. 2 ⊢ (φ → A ∈ B)
5		nfv 1619	. . 3 ⊢ Ⅎxχ
6		nfcv 2490	. . 3 ⊢ ℲxA
7	5, 6	spcimgft 2931	. 2 ⊢ (∀x(x = A → (ψ → χ)) → (A ∈ B → (∀xψ → χ)))
8	3, 4, 7	sylc 56	1 ⊢ (φ → (∀xψ → χ))

Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710

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

This theorem is referenced by:  spcdv  2938  spcimedv  2939  rspcimdv  2957