Theorem rabtru

Description: Abstract builder using the constant wff T. . (Contributed by Thierry Arnoux, 4-May-2020)

	Ref	Expression
Hypothesis	rabtru.1	$$⊢\underset{\_}{Ⅎ}xA$$
Assertion	rabtru	$$⊢\left\{x\in A|\top \right\}=A$$

Proof

Step	Hyp	Ref	Expression
1		rabtru.1	$$⊢\underset{\_}{Ⅎ}xA$$
2		tru	$$⊢\top$$
3		nfcv	$$⊢\underset{\_}{Ⅎ}xy$$
4		nftru	$$⊢Ⅎx\top$$
5		biidd	$$⊢x=y\to \left(\top \leftrightarrow \top \right)$$
6	3 1 4 5	elrabf	$$⊢y\in \left\{x\in A|\top \right\}\leftrightarrow \left(y\in A\wedge \top \right)$$
7	2 6	mpbiran2	$$⊢y\in \left\{x\in A|\top \right\}\leftrightarrow y\in A$$
8	7	eqriv	$$⊢\left\{x\in A|\top \right\}=A$$