Theorem frege105

Description: Proposition 105 of Frege1879 p. 73. (Contributed by RP, 7-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	frege103.z	$$⊢Z\in V$$
Assertion	frege105	$$⊢\left(\neg X\mathrm{t+}\left(R\right)Z\to Z=X\right)\to X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Z$$

Proof

Step	Hyp	Ref	Expression
1		frege103.z	$$⊢Z\in V$$
2	1	dffrege99	$$⊢\left(\neg X\mathrm{t+}\left(R\right)Z\to Z=X\right)\leftrightarrow X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Z$$
3		frege52aid	$$⊢\left(\left(\neg X\mathrm{t+}\left(R\right)Z\to Z=X\right)\leftrightarrow X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Z\right)\to \left(\left(\neg X\mathrm{t+}\left(R\right)Z\to Z=X\right)\to X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Z\right)$$
4	2 3	ax-mp	$$⊢\left(\neg X\mathrm{t+}\left(R\right)Z\to Z=X\right)\to X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Z$$