Theorem frege104

Description: Proposition 104 of Frege1879 p. 73.

Note: in the Bauer-Meenfelberg translation published in van Heijenoort's collectionFrom Frege to Goedel, this proof has the minor clause and result swapped. (Contributed by RP, 7-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	frege103.z	$$⊢Z\in V$$
Assertion	frege104	$$⊢X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Z\to \left(\neg X\mathrm{t+}\left(R\right)Z\to X=Z\right)$$

Proof

Step	Hyp	Ref	Expression
1		frege103.z	$$⊢Z\in V$$
2	1	elexi	$$⊢Z\in \mathrm{V}$$
3		eqeq1	$$⊢z=Z\to \left(z=X\leftrightarrow Z=X\right)$$
4		eqeq2	$$⊢z=Z\to \left(X=z\leftrightarrow X=Z\right)$$
5	3 4	imbi12d	$$⊢z=Z\to \left(\left(z=X\to X=z\right)\leftrightarrow \left(Z=X\to X=Z\right)\right)$$
6		frege55c	$$⊢z=X\to X=z$$
7	2 5 6	vtocl	$$⊢Z=X\to X=Z$$
8	1	frege103	$$⊢\left(Z=X\to X=Z\right)\to \left(X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Z\to \left(\neg X\mathrm{t+}\left(R\right)Z\to X=Z\right)\right)$$
9	7 8	ax-mp	$$⊢X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Z\to \left(\neg X\mathrm{t+}\left(R\right)Z\to X=Z\right)$$