Theorem dedhb 3675

Description: A deduction theorem for converting the inference ⊢ Ⅎ𝑥𝐴 => ⊢ 𝜑 into a closed theorem. Use nfa1 2192 and nfab 2937 to eliminate the hypothesis of the substitution instance 𝜓 of the inference. For converting the inference form into a deduction form, abidnf 3674 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 3674	. . . 4 ⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴)
3	2	eqcomd 2775	. . 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)

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1565   = wceq 1567   ∈ wcel 2149  {cab 2747  Ⅎwnfc 2916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918

This theorem is referenced by: (None)