Theorem subtr2 37318

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

Hypotheses

Ref	Expression
subtr.1	⊢ Ⅎ𝑥𝐴
subtr.2	⊢ Ⅎ𝑥𝐵
subtr2.3	⊢ Ⅎ𝑥𝜓
subtr2.4	⊢ Ⅎ𝑥𝜒
subtr2.5	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
subtr2.6	⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))

Assertion

Ref	Expression
subtr2	⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 = 𝐵 → (𝜓 ↔ 𝜒)))

Proof of Theorem subtr2

Step	Hyp	Ref	Expression
1		subtr.1	. . 3 ⊢ Ⅎ𝑥𝐴
2		subtr.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
3	1, 2	nfeq 2939	. . . 4 ⊢ Ⅎ𝑥 𝐴 = 𝐵
4		subtr2.3	. . . . 5 ⊢ Ⅎ𝑥𝜓
5		subtr2.4	. . . . 5 ⊢ Ⅎ𝑥𝜒
6	4, 5	nfbi 1938	. . . 4 ⊢ Ⅎ𝑥(𝜓 ↔ 𝜒)
7	3, 6	nfim 1931	. . 3 ⊢ Ⅎ𝑥(𝐴 = 𝐵 → (𝜓 ↔ 𝜒))
8		eqeq1 2767	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐵 ↔ 𝐴 = 𝐵))
9		subtr2.5	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
10	9	bibi1d 346	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜒)))
11	8, 10	imbi12d 347	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) ↔ (𝐴 = 𝐵 → (𝜓 ↔ 𝜒))))
12		subtr2.6	. . 3 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))
13	1, 7, 11, 12	vtoclgf 3533	. 2 ⊢ (𝐴 ∈ 𝐶 → (𝐴 = 𝐵 → (𝜓 ↔ 𝜒)))
14	13	adantr 486	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 = 𝐵 → (𝜓 ↔ 𝜒)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  Ⅎwnf 1816   ∈ wcel 2147  Ⅎwnfc 2911

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-v 3456

This theorem is used by: (None)