Theorem subtr

Description: Transitivity of implicit substitution. (Contributed by Jeff Hankins, 13-Sep-2009) (Proof shortened by Mario Carneiro, 11-Dec-2016)

	Ref	Expression
Hypotheses	subtr.1	$$⊢\underset{\_}{Ⅎ}xA$$
	subtr.2	$$⊢\underset{\_}{Ⅎ}xB$$
	subtr.3	$$⊢\underset{\_}{Ⅎ}xY$$
	subtr.4	$$⊢\underset{\_}{Ⅎ}xZ$$
	subtr.5	$$⊢x=A\to X=Y$$
	subtr.6	$$⊢x=B\to X=Z$$
Assertion	subtr	$$⊢\left(A\in C\wedge B\in D\right)\to \left(A=B\to Y=Z\right)$$

Proof

Step	Hyp	Ref	Expression
1		subtr.1	$$⊢\underset{\_}{Ⅎ}xA$$
2		subtr.2	$$⊢\underset{\_}{Ⅎ}xB$$
3		subtr.3	$$⊢\underset{\_}{Ⅎ}xY$$
4		subtr.4	$$⊢\underset{\_}{Ⅎ}xZ$$
5		subtr.5	$$⊢x=A\to X=Y$$
6		subtr.6	$$⊢x=B\to X=Z$$
7	1 2	nfeq	$$⊢ℲxA=B$$
8	3 4	nfeq	$$⊢ℲxY=Z$$
9	7 8	nfim	$$⊢Ⅎx\left(A=B\to Y=Z\right)$$
10		eqeq1	$$⊢x=A\to \left(x=B\leftrightarrow A=B\right)$$
11	5	eqeq1d	$$⊢x=A\to \left(X=Z\leftrightarrow Y=Z\right)$$
12	10 11	imbi12d	$$⊢x=A\to \left(\left(x=B\to X=Z\right)\leftrightarrow \left(A=B\to Y=Z\right)\right)$$
13	1 9 12 6	vtoclgf	$$⊢A\in C\to \left(A=B\to Y=Z\right)$$
14	13	adantr	$$⊢\left(A\in C\wedge B\in D\right)\to \left(A=B\to Y=Z\right)$$