Theorem cla4egf 1395

Description: Existential specialization with implicit substitution.

Hypotheses

Ref	Expression
cla4gf.1	⊢ (y ∈ A → ∀x y ∈ A)
cla4gf.2	⊢ (ψ → ∀xψ)
cla4gf.3	⊢ (x = A → (φ ↔ ψ))

Assertion

Ref	Expression
cla4egf	⊢ (A ∈ B → (ψ → ∃xφ))

Distinct variable group(s):   x,y   y,A

Proof of Theorem cla4egf

Step	Hyp	Ref	Expression
1		cla4gf.1	. . . 4 ⊢ (y ∈ A → ∀x y ∈ A)
2		cla4gf.2	. . . . 5 ⊢ (ψ → ∀xψ)
3	2	hbne 699	. . . 4 ⊢ (¬ ψ → ∀x ¬ ψ)
4		cla4gf.3	. . . . 5 ⊢ (x = A → (φ ↔ ψ))
5	4	negbid 463	. . . 4 ⊢ (x = A → (¬ φ ↔ ¬ ψ))
6	1, 3, 5	cla4gf 1394	. . 3 ⊢ (A ∈ B → (∀x ¬ φ → ¬ ψ))
7	6	con2d 83	. 2 ⊢ (A ∈ B → (ψ → ¬ ∀x ¬ φ))
8		df-ex 679	. 2 ⊢ (∃xφ ↔ ¬ ∀x ¬ φ)
9	7, 8	syl6ibr 186	1 ⊢ (A ∈ B → (ψ → ∃xφ))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127  ∀wal 672  ∃wex 678   = wceq 1091   ∈ wcel 1092

This theorem is referenced by:  cla4egv 1397  onminex 2275  zfrep6 2744

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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