Theorem dedhb 3376

Description: A deduction theorem for converting the inference ⊢ Ⅎ𝑥𝐴 => ⊢ 𝜑 into a closed theorem. Use nfa1 2028 and nfab 2769 to eliminate the hypothesis of the substitution instance 𝜓 of the inference. For converting the inference form into a deduction form, abidnf 3375 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 3375	. . . 4 ⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴)
3	2	eqcomd 2628	. . 3 ⊢ (Ⅎ𝑥𝐴 → 𝐴 = {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴})
4		dedhb.1	. . 3 ⊢ (𝐴 = {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} → (𝜑 ↔ 𝜓))
5	3, 4	syl 17	. 2 ⊢ (Ⅎ𝑥𝐴 → (𝜑 ↔ 𝜓))
6	1, 5	mpbiri 248	1 ⊢ (Ⅎ𝑥𝐴 → 𝜑)

Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1481   = wceq 1483   ∈ wcel 1990  {cab 2608  Ⅎwnfc 2751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753

This theorem is referenced by: (None)