Theorem cla4gf 1394

Description: Rule of specialization with implicit substitution. Compare Theorem 7.3 of [Quine] p. 44.

Hypotheses

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

Assertion

Ref	Expression
cla4gf	⊢ (A ∈ B → (∀xφ → ψ))

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

Proof of Theorem cla4gf

Step	Hyp	Ref	Expression
1		elisset 1354	. 2 ⊢ (A ∈ B → A ∈ V)
2		isset 1351	. . . . 5 ⊢ (A ∈ V ↔ ∃y y = A)
3		cla4gf.1	. . . . . . 7 ⊢ (y ∈ A → ∀x y ∈ A)
4	3	hbeleq 1173	. . . . . 6 ⊢ (y = A → ∀x y = A)
5		ax-17 925	. . . . . 6 ⊢ (x = A → ∀y x = A)
6		cleq1 1107	. . . . . 6 ⊢ (y = x → (y = A ↔ x = A))
7	4, 5, 6	cbvex 849	. . . . 5 ⊢ (∃y y = A ↔ ∃x x = A)
8	2, 7	bitr 151	. . . 4 ⊢ (A ∈ V ↔ ∃x x = A)
9		cla4gf.3	. . . . . 6 ⊢ (x = A → (φ ↔ ψ))
10	9	biimpd 135	. . . . 5 ⊢ (x = A → (φ → ψ))
11	10	19.22i 723	. . . 4 ⊢ (∃x x = A → ∃x(φ → ψ))
12	8, 11	sylbi 174	. . 3 ⊢ (A ∈ V → ∃x(φ → ψ))
13		cla4gf.2	. . . 4 ⊢ (ψ → ∀xψ)
14	13	19.36 757	. . 3 ⊢ (∃x(φ → ψ) ↔ (∀xφ → ψ))
15	12, 14	sylib 173	. 2 ⊢ (A ∈ V → (∀xφ → ψ))
16	1, 15	syl 12	1 ⊢ (A ∈ B → (∀xφ → ψ))

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672  ∃wex 678   = wceq 1091   ∈ wcel 1092  Vcvv 1348

This theorem is referenced by:  cla4egf 1395  cla4gv 1396

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

metamath.org