Theorem bnj91

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj91.1	$$⊢\phi \leftrightarrow f\left(\varnothing \right)=pred\left(x,A,R\right)$$
	bnj91.2	$$⊢Z\in \mathrm{V}$$
Assertion	bnj91	$$⊢\underset{\dot{}}{[}Z/y\underset{\dot{}}{]}\phi \leftrightarrow f\left(\varnothing \right)=pred\left(x,A,R\right)$$

Proof

Step	Hyp	Ref	Expression
1		bnj91.1	$$⊢\phi \leftrightarrow f\left(\varnothing \right)=pred\left(x,A,R\right)$$
2		bnj91.2	$$⊢Z\in \mathrm{V}$$
3	1	sbcbii	$$⊢\underset{\dot{}}{[}Z/y\underset{\dot{}}{]}\phi \leftrightarrow \underset{\dot{}}{[}Z/y\underset{\dot{}}{]}f\left(\varnothing \right)=pred\left(x,A,R\right)$$
4	2	bnj525	$$⊢\underset{\dot{}}{[}Z/y\underset{\dot{}}{]}f\left(\varnothing \right)=pred\left(x,A,R\right)\leftrightarrow f\left(\varnothing \right)=pred\left(x,A,R\right)$$
5	3 4	bitri	$$⊢\underset{\dot{}}{[}Z/y\underset{\dot{}}{]}\phi \leftrightarrow f\left(\varnothing \right)=pred\left(x,A,R\right)$$