Theorem subtr2

Description: Transitivity of implicit substitution into a wff. (Contributed by Jeff Hankins, 19-Sep-2009) (Proof shortened by Mario Carneiro, 11-Dec-2016)

	Ref	Expression
Hypotheses	subtr.1	$$⊢\underset{\_}{Ⅎ}xA$$
	subtr.2	$$⊢\underset{\_}{Ⅎ}xB$$
	subtr2.3	$$⊢Ⅎx\psi$$
	subtr2.4	$$⊢Ⅎx\chi$$
	subtr2.5	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
	subtr2.6	$$⊢x=B\to \left(\phi \leftrightarrow \chi \right)$$
Assertion	subtr2	$$⊢\left(A\in C\wedge B\in D\right)\to \left(A=B\to \left(\psi \leftrightarrow \chi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		subtr.1	$$⊢\underset{\_}{Ⅎ}xA$$
2		subtr.2	$$⊢\underset{\_}{Ⅎ}xB$$
3		subtr2.3	$$⊢Ⅎx\psi$$
4		subtr2.4	$$⊢Ⅎx\chi$$
5		subtr2.5	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
6		subtr2.6	$$⊢x=B\to \left(\phi \leftrightarrow \chi \right)$$
7	1 2	nfeq	$$⊢ℲxA=B$$
8	3 4	nfbi	$$⊢Ⅎx\left(\psi \leftrightarrow \chi \right)$$
9	7 8	nfim	$$⊢Ⅎx\left(A=B\to \left(\psi \leftrightarrow \chi \right)\right)$$
10		eqeq1	$$⊢x=A\to \left(x=B\leftrightarrow A=B\right)$$
11	5	bibi1d	$$⊢x=A\to \left(\left(\phi \leftrightarrow \chi \right)\leftrightarrow \left(\psi \leftrightarrow \chi \right)\right)$$
12	10 11	imbi12d	$$⊢x=A\to \left(\left(x=B\to \left(\phi \leftrightarrow \chi \right)\right)\leftrightarrow \left(A=B\to \left(\psi \leftrightarrow \chi \right)\right)\right)$$
13	1 9 12 6	vtoclgf	$$⊢A\in C\to \left(A=B\to \left(\psi \leftrightarrow \chi \right)\right)$$
14	13	adantr	$$⊢\left(A\in C\wedge B\in D\right)\to \left(A=B\to \left(\psi \leftrightarrow \chi \right)\right)$$