Theorem dedhb 3007

Description: A deduction theorem for converting the inference ⊢ ℲxA => ⊢ φ into a closed theorem. Use nfa1 1788 and nfab 2494 to eliminate the hypothesis of the substitution instance ψ of the inference. For converting the inference form into a deduction form, abidnf 3006 is useful. (Contributed by NM, 8-Dec-2006.)

Hypotheses

Ref	Expression
dedhb.1	⊢ (A = {z ∣ ∀x z ∈ A} → (φ ↔ ψ))
dedhb.2	⊢ ψ

Assertion

Ref	Expression
dedhb	⊢ (ℲxA → φ)

Distinct variable groups:   x,z   z,A

Allowed substitution hints:   φ(x,z)   ψ(x,z)   A(x)

Proof of Theorem dedhb

Step	Hyp	Ref	Expression
1		dedhb.2	. 2 ⊢ ψ
2		abidnf 3006	. . . 4 ⊢ (ℲxA → {z ∣ ∀x z ∈ A} = A)
3	2	eqcomd 2358	. . 3 ⊢ (ℲxA → A = {z ∣ ∀x z ∈ A})
4		dedhb.1	. . 3 ⊢ (A = {z ∣ ∀x z ∈ A} → (φ ↔ ψ))
5	3, 4	syl 15	. 2 ⊢ (ℲxA → (φ ↔ ψ))
6	1, 5	mpbiri 224	1 ⊢ (ℲxA → φ)

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2477

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-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

This theorem is referenced by: (None)