Theorem dedhb 3664

Description: A deduction theorem for converting the inference ⊢ Ⅎ𝑥𝐴 => ⊢ 𝜑 into a closed theorem. Use nfa1 2191 and nfab 2932 to eliminate the hypothesis of the substitution instance 𝜓 of the inference. For converting the inference form into a deduction form, abidnf 3663 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 3663	. . . 4 ⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴)
3	2	eqcomd 2769	. . 3 ⊢ (Ⅎ𝑥𝐴 → 𝐴 = {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴})
4		dedhb.1	. . 3 ⊢ (𝐴 = {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} → (𝜑 ↔ 𝜓))
5	3, 4	syl 18	. 2 ⊢ (Ⅎ𝑥𝐴 → (𝜑 ↔ 𝜓))
6	1, 5	mpbiri 261	1 ⊢ (Ⅎ𝑥𝐴 → 𝜑)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568   = wceq 1570   ∈ wcel 2147  {cab 2741  Ⅎwnfc 2911

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913

This theorem is used by: (None)