Theorem subtr 34503

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

Hypotheses

Ref	Expression
subtr.1	⊢ Ⅎ𝑥𝐴
subtr.2	⊢ Ⅎ𝑥𝐵
subtr.3	⊢ Ⅎ𝑥𝑌
subtr.4	⊢ Ⅎ𝑥𝑍
subtr.5	⊢ (𝑥 = 𝐴 → 𝑋 = 𝑌)
subtr.6	⊢ (𝑥 = 𝐵 → 𝑋 = 𝑍)

Assertion

Ref	Expression
subtr	⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 = 𝐵 → 𝑌 = 𝑍))

Proof of Theorem subtr

Step	Hyp	Ref	Expression
1		subtr.1	. . 3 ⊢ Ⅎ𝑥𝐴
2		subtr.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
3	1, 2	nfeq 2920	. . . 4 ⊢ Ⅎ𝑥 𝐴 = 𝐵
4		subtr.3	. . . . 5 ⊢ Ⅎ𝑥𝑌
5		subtr.4	. . . . 5 ⊢ Ⅎ𝑥𝑍
6	4, 5	nfeq 2920	. . . 4 ⊢ Ⅎ𝑥 𝑌 = 𝑍
7	3, 6	nfim 1899	. . 3 ⊢ Ⅎ𝑥(𝐴 = 𝐵 → 𝑌 = 𝑍)
8		eqeq1 2742	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐵 ↔ 𝐴 = 𝐵))
9		subtr.5	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑋 = 𝑌)
10	9	eqeq1d 2740	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑋 = 𝑍 ↔ 𝑌 = 𝑍))
11	8, 10	imbi12d 345	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 = 𝐵 → 𝑋 = 𝑍) ↔ (𝐴 = 𝐵 → 𝑌 = 𝑍)))
12		subtr.6	. . 3 ⊢ (𝑥 = 𝐵 → 𝑋 = 𝑍)
13	1, 7, 11, 12	vtoclgf 3503	. 2 ⊢ (𝐴 ∈ 𝐶 → (𝐴 = 𝐵 → 𝑌 = 𝑍))
14	13	adantr 481	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 = 𝐵 → 𝑌 = 𝑍))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Ⅎwnfc 2887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-v 3434

This theorem is referenced by: (None)