Theorem dedhb 2920

Description: A deduction theorem for converting the inference ⊢ Ⅎ𝑥𝐴 => ⊢ 𝜑 into a closed theorem. Use nfa1 1551 and nfab 2336 to eliminate the hypothesis of the substitution instance 𝜓 of the inference. For converting the inference form into a deduction form, abidnf 2919 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 2919	. . . 4 ⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴)
3	2	eqcomd 2194	. . 3 ⊢ (Ⅎ𝑥𝐴 → 𝐴 = {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴})
4		dedhb.1	. . 3 ⊢ (𝐴 = {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} → (𝜑 ↔ 𝜓))
5	3, 4	syl 14	. 2 ⊢ (Ⅎ𝑥𝐴 → (𝜑 ↔ 𝜓))
6	1, 5	mpbiri 168	1 ⊢ (Ⅎ𝑥𝐴 → 𝜑)

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1361   = wceq 1363   ∈ wcel 2159  {cab 2174  Ⅎwnfc 2318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-11 1516  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170

This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320

This theorem is referenced by: (None)