Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Proof revision is marked as discouraged because the minimizer replaces albidv with dral2 , leading to a one byte longer proof. However feel free to manually edit it according to conventions. (TODO: dral2 depends on ax-13 , hence its usage during minimizing is discouraged. Check in the long run whether this is a permanent restriction). Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by Mario Carneiro, 8-Oct-2016) Avoid ax-8 . (Revised by Wolf Lammen, 22-Sep-2024) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | drnfc1.1 | |- ( A. x x = y -> A = B ) |
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| Assertion | drnfc2 | |- ( A. x x = y -> ( F/_ z A <-> F/_ z B ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | drnfc1.1 | |- ( A. x x = y -> A = B ) |
|
| 2 | eleq2w2 | |- ( A = B -> ( w e. A <-> w e. B ) ) |
|
| 3 | 1 2 | syl | |- ( A. x x = y -> ( w e. A <-> w e. B ) ) |
| 4 | 3 | drnf2 | |- ( A. x x = y -> ( F/ z w e. A <-> F/ z w e. B ) ) |
| 5 | 4 | albidv | |- ( A. x x = y -> ( A. w F/ z w e. A <-> A. w F/ z w e. B ) ) |
| 6 | df-nfc | |- ( F/_ z A <-> A. w F/ z w e. A ) |
|
| 7 | df-nfc | |- ( F/_ z B <-> A. w F/ z w e. B ) |
|
| 8 | 5 6 7 | 3bitr4g | |- ( A. x x = y -> ( F/_ z A <-> F/_ z B ) ) |