Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > dedhb | Structured version Visualization version GIF version |
Description: A deduction theorem for converting the inference ⊢ Ⅎ𝑥𝐴 => ⊢ 𝜑 into a closed theorem. Use nfa1 2149 and nfab 2911 to eliminate the hypothesis of the substitution instance 𝜓 of the inference. For converting the inference form into a deduction form, abidnf 3638 is useful. (Contributed by NM, 8-Dec-2006.) |
Ref | Expression |
---|---|
dedhb.1 | ⊢ (𝐴 = {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} → (𝜑 ↔ 𝜓)) |
dedhb.2 | ⊢ 𝜓 |
Ref | Expression |
---|---|
dedhb | ⊢ (Ⅎ𝑥𝐴 → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dedhb.2 | . 2 ⊢ 𝜓 | |
2 | abidnf 3638 | . . . 4 ⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴) | |
3 | 2 | eqcomd 2743 | . . 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 257 | 1 ⊢ (Ⅎ𝑥𝐴 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1537 = wceq 1539 ∈ wcel 2107 {cab 2714 Ⅎwnfc 2885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-12 2172 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1784 df-nf 1788 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |