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Mirrors > Home > MPE Home > Th. List > dral2 | Structured version Visualization version GIF version |
Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). Usage of this theorem is discouraged because it depends on ax-13 2390. Usage of albidv 1921 is preferred, which requires fewer axioms. (Contributed by NM, 27-Feb-2005.) Allow a shortening of dral1 2461. (Revised by Wolf Lammen, 4-Mar-2018.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dral1.1 | ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
dral2 | ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfae 2455 | . 2 ⊢ Ⅎ𝑧∀𝑥 𝑥 = 𝑦 | |
2 | dral1.1 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | 1, 2 | albid 2224 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2145 ax-11 2161 ax-12 2177 ax-13 2390 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 |
This theorem is referenced by: dral1ALT 2462 sbal1 2572 sbal2 2573 sbal2OLD 2574 drnfc1OLD 3000 axpownd 10025 wl-sbalnae 34800 |
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