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Mirrors > Home > MPE Home > Th. List > drnf2 | Structured version Visualization version GIF version |
Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Usage of this theorem is discouraged because it depends on ax-13 2371. Usage of nfbidv 1928 is preferred, which requires fewer axioms. (Contributed by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 5-May-2018.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dral1.1 | ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
drnf2 | ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 ↔ Ⅎ𝑧𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfae 2432 | . 2 ⊢ Ⅎ𝑧∀𝑥 𝑥 = 𝑦 | |
2 | dral1.1 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | 1, 2 | nfbidf 2225 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 ↔ Ⅎ𝑧𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1540 Ⅎwnf 1790 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-10 2144 ax-11 2161 ax-12 2178 ax-13 2371 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-tru 1545 df-ex 1787 df-nf 1791 |
This theorem is referenced by: nfsb4t 2503 drnfc2 2920 drnfc2OLD 2921 |
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