Theorem imaeqsalv

Description: Substitute a function value into a universal quantifier over an image. (Contributed by Scott Fenton, 27-Sep-2024)

	Ref	Expression
Hypothesis	imaeqsex.1	$$⊢x=F\left(y\right)\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	imaeqsalv	$$⊢\left(FFnA\wedge B\subseteq A\right)\to \left(\forall x\in F\left[B\right]\phi \leftrightarrow \forall y\in B\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		imaeqsex.1	$$⊢x=F\left(y\right)\to \left(\phi \leftrightarrow \psi \right)$$
2	1	notbid	$$⊢x=F\left(y\right)\to \left(\neg \phi \leftrightarrow \neg \psi \right)$$
3	2	imaeqsexv	$$⊢\left(FFnA\wedge B\subseteq A\right)\to \left(\exists x\in F\left[B\right]\neg \phi \leftrightarrow \exists y\in B\neg \psi \right)$$
4	3	notbid	$$⊢\left(FFnA\wedge B\subseteq A\right)\to \left(\neg \exists x\in F\left[B\right]\neg \phi \leftrightarrow \neg \exists y\in B\neg \psi \right)$$
5		dfral2	$$⊢\forall x\in F\left[B\right]\phi \leftrightarrow \neg \exists x\in F\left[B\right]\neg \phi$$
6		dfral2	$$⊢\forall y\in B\psi \leftrightarrow \neg \exists y\in B\neg \psi$$
7	4 5 6	3bitr4g	$$⊢\left(FFnA\wedge B\subseteq A\right)\to \left(\forall x\in F\left[B\right]\phi \leftrightarrow \forall y\in B\psi \right)$$