Theorem sb3b

Description: Simplified definition of substitution when variables are distinct. This is the biconditional strengthening of sb3 . Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by BJ, 6-Oct-2018) Shorten sb3 . (Revised by Wolf Lammen, 21-Feb-2021) (New usage is discouraged.)

	Ref	Expression
Assertion	sb3b	$$⊢\neg \forall xx=y\to \left(\left[y/x\right]\phi \leftrightarrow \exists x\left(x=y\wedge \phi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		sb4b	$$⊢\neg \forall xx=y\to \left(\left[y/x\right]\phi \leftrightarrow \forall x\left(x=y\to \phi \right)\right)$$
2		equs5	$$⊢\neg \forall xx=y\to \left(\exists x\left(x=y\wedge \phi \right)\leftrightarrow \forall x\left(x=y\to \phi \right)\right)$$
3	1 2	bitr4d	$$⊢\neg \forall xx=y\to \left(\left[y/x\right]\phi \leftrightarrow \exists x\left(x=y\wedge \phi \right)\right)$$