Theorem wl-dral1d

Description: A version of dral1 with a context. Note: At first glance one might be tempted to generalize this (or a similar) theorem by weakening the first two hypotheses adding a x = y , A. x x = y or ph antecedent. wl-equsal1i and nf5di show that this is in fact pointless. (Contributed by Wolf Lammen, 28-Jul-2019)

	Ref	Expression
Hypotheses	wl-dral1d.1	$$⊢Ⅎx\phi$$
	wl-dral1d.2	$$⊢Ⅎy\phi$$
	wl-dral1d.3	$$⊢\phi \to \left(x=y\to \left(\psi \leftrightarrow \chi \right)\right)$$
Assertion	wl-dral1d	$$⊢\phi \to \left(\forall xx=y\to \left(\forall x\psi \leftrightarrow \forall y\chi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		wl-dral1d.1	$$⊢Ⅎx\phi$$
2		wl-dral1d.2	$$⊢Ⅎy\phi$$
3		wl-dral1d.3	$$⊢\phi \to \left(x=y\to \left(\psi \leftrightarrow \chi \right)\right)$$
4	3	com12	$$⊢x=y\to \left(\phi \to \left(\psi \leftrightarrow \chi \right)\right)$$
5	4	pm5.74d	$$⊢x=y\to \left(\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \chi \right)\right)$$
6	5	sps	$$⊢\forall xx=y\to \left(\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \chi \right)\right)$$
7	6	dral1	$$⊢\forall xx=y\to \left(\forall x\left(\phi \to \psi \right)\leftrightarrow \forall y\left(\phi \to \chi \right)\right)$$
8	1	19.21	$$⊢\forall x\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \forall x\psi \right)$$
9	2	19.21	$$⊢\forall y\left(\phi \to \chi \right)\leftrightarrow \left(\phi \to \forall y\chi \right)$$
10	7 8 9	3bitr3g	$$⊢\forall xx=y\to \left(\left(\phi \to \forall x\psi \right)\leftrightarrow \left(\phi \to \forall y\chi \right)\right)$$
11	10	pm5.74rd	$$⊢\forall xx=y\to \left(\phi \to \left(\forall x\psi \leftrightarrow \forall y\chi \right)\right)$$
12	11	com12	$$⊢\phi \to \left(\forall xx=y\to \left(\forall x\psi \leftrightarrow \forall y\chi \right)\right)$$