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Mirrors > Home > MPE Home > Th. List > nfbidfOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of nfbidf 2092 as of 6-Oct-2021. (Contributed by Mario Carneiro, 4-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nfbidfOLD.1 | ⊢ Ⅎ𝑥𝜑 |
nfbidfOLD.2 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
nfbidfOLD | ⊢ (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfbidfOLD.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | nfbidfOLD.2 | . . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
3 | 1, 2 | albidOLD 2199 | . . . 4 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒)) |
4 | 2, 3 | imbi12d 334 | . . 3 ⊢ (𝜑 → ((𝜓 → ∀𝑥𝜓) ↔ (𝜒 → ∀𝑥𝜒))) |
5 | 1, 4 | albidOLD 2199 | . 2 ⊢ (𝜑 → (∀𝑥(𝜓 → ∀𝑥𝜓) ↔ ∀𝑥(𝜒 → ∀𝑥𝜒))) |
6 | df-nfOLD 1721 | . 2 ⊢ (Ⅎ𝑥𝜓 ↔ ∀𝑥(𝜓 → ∀𝑥𝜓)) | |
7 | df-nfOLD 1721 | . 2 ⊢ (Ⅎ𝑥𝜒 ↔ ∀𝑥(𝜒 → ∀𝑥𝜒)) | |
8 | 5, 6, 7 | 3bitr4g 303 | 1 ⊢ (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∀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-ex 1705 df-nfOLD 1721 |
This theorem is referenced by: (None) |
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