Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > nfan1OLD | Structured version Visualization version GIF version |
Description: Obsolete proof of nfan1 2068 as of 6-Oct-2021. (Contributed by Mario Carneiro, 3-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nfan1OLD.1 | ⊢ Ⅎ𝑥𝜑 |
nfan1OLD.2 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfan1OLD | ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfan1OLD.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
2 | 1 | nfrdOLD 2190 | . . . 4 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) |
3 | 2 | imdistani 726 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝜑 ∧ ∀𝑥𝜓)) |
4 | nfan1OLD.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
5 | 4 | 19.28OLD 2235 | . . 3 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓)) |
6 | 3, 5 | sylibr 224 | . 2 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑥(𝜑 ∧ 𝜓)) |
7 | 6 | nfiOLD 1734 | 1 ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∀wal 1481 ℲwnfOLD 1709 |
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-12 2047 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 df-nfOLD 1721 |
This theorem is referenced by: nfanOLDOLD 2237 |
Copyright terms: Public domain | W3C validator |