Theorem ralxfr 5380

Description: Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.)

Hypotheses

Ref	Expression
ralxfr.1	⊢ (𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵)
ralxfr.2	⊢ (𝑥 ∈ 𝐵 → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)
ralxfr.3	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
ralxfr	⊢ (∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐶 𝜓)

Distinct variable groups:   𝜓,𝑥   𝜑,𝑦   𝑥,𝐴   𝑥,𝑦,𝐵   𝑥,𝐶

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝐴(𝑦)   𝐶(𝑦)

Proof of Theorem ralxfr

Step	Hyp	Ref	Expression
1		ralxfr.1	. . . 4 ⊢ (𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵)
2	1	adantl 487	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵)
3		ralxfr.2	. . . 4 ⊢ (𝑥 ∈ 𝐵 → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)
4	3	adantl 487	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)
5		ralxfr.3	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
6	5	adantl 487	. . 3 ⊢ ((⊤ ∧ 𝑥 = 𝐴) → (𝜑 ↔ 𝜓))
7	2, 4, 6	ralxfrd 5374	. 2 ⊢ (⊤ → (∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐶 𝜓))
8	7	mptru 1577	1 ⊢ (∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐶 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  ⊤wtru 1571   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090

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-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091

This theorem is used by:  rexxfr  5382  infm3  12303