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Mirrors > Home > MPE Home > Th. List > nfbiiOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of nfbii 1770 as of 6-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nfbiiOLD.1 | ⊢ (𝜑 ↔ 𝜓) |
Ref | Expression |
---|---|
nfbiiOLD | ⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfbiiOLD.1 | . . . 4 ⊢ (𝜑 ↔ 𝜓) | |
2 | 1 | albii 1737 | . . . 4 ⊢ (∀𝑥𝜑 ↔ ∀𝑥𝜓) |
3 | 1, 2 | imbi12i 339 | . . 3 ⊢ ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑥𝜓)) |
4 | 3 | albii 1737 | . 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥(𝜓 → ∀𝑥𝜓)) |
5 | df-nfOLD 1712 | . 2 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑)) | |
6 | df-nfOLD 1712 | . 2 ⊢ (Ⅎ𝑥𝜓 ↔ ∀𝑥(𝜓 → ∀𝑥𝜓)) | |
7 | 4, 5, 6 | 3bitr4i 291 | 1 ⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 ℲwnfOLD 1700 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 |
This theorem depends on definitions: df-bi 196 df-nfOLD 1712 |
This theorem is referenced by: nfxfrOLD 1825 nfxfrdOLD 1826 |
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