Theorem frege55lem1c

Description: Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019)

	Ref	Expression
Assertion	frege55lem1c	$$⊢\left(\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}x=B\right)\to \left(\phi \to A=B\right)$$

Proof

Step	Hyp	Ref	Expression
1		df-sbc	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}x=B\leftrightarrow A\in \left\{x|x=B\right\}$$
2		eqeq1	$$⊢x=A\to \left(x=B\leftrightarrow A=B\right)$$
3	2	elabg	$$⊢A\in \left\{x|x=B\right\}\to \left(A\in \left\{x|x=B\right\}\leftrightarrow A=B\right)$$
4	3	ibi	$$⊢A\in \left\{x|x=B\right\}\to A=B$$
5	1 4	sylbi	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}x=B\to A=B$$
6	5	imim2i	$$⊢\left(\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}x=B\right)\to \left(\phi \to A=B\right)$$