Theorem frege127

Description: Communte antecedents of frege126 . Proposition 127 of Frege1879 p. 82. (Contributed by RP, 9-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege123.x	$$⊢X\in U$$
	frege123.y	$$⊢Y\in V$$
	frege124.m	$$⊢M\in W$$
	frege124.r	$$⊢R\in S$$
Assertion	frege127	$$⊢\mathrm{Fun}{R^{-1}}^{-1}\to \left(Y\mathrm{t+}\left(R\right)M\to \left(YRX\to \left(\neg X\mathrm{t+}\left(R\right)M\to M\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)X\right)\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		frege123.x	$$⊢X\in U$$
2		frege123.y	$$⊢Y\in V$$
3		frege124.m	$$⊢M\in W$$
4		frege124.r	$$⊢R\in S$$
5	1 2 3 4	frege126	$$⊢\mathrm{Fun}{R^{-1}}^{-1}\to \left(YRX\to \left(Y\mathrm{t+}\left(R\right)M\to \left(\neg X\mathrm{t+}\left(R\right)M\to M\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)X\right)\right)\right)$$
6		frege12	$$⊢\left(\mathrm{Fun}{R^{-1}}^{-1}\to \left(YRX\to \left(Y\mathrm{t+}\left(R\right)M\to \left(\neg X\mathrm{t+}\left(R\right)M\to M\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)X\right)\right)\right)\right)\to \left(\mathrm{Fun}{R^{-1}}^{-1}\to \left(Y\mathrm{t+}\left(R\right)M\to \left(YRX\to \left(\neg X\mathrm{t+}\left(R\right)M\to M\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)X\right)\right)\right)\right)$$
7	5 6	ax-mp	$$⊢\mathrm{Fun}{R^{-1}}^{-1}\to \left(Y\mathrm{t+}\left(R\right)M\to \left(YRX\to \left(\neg X\mathrm{t+}\left(R\right)M\to M\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)X\right)\right)\right)$$