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Metamath Proof Explorer

Theorem ralxfrd

Description: Transfer universal quantification from a variable x to another variable y contained in expression A . (Contributed by NM, 15-Aug-2014) (Proof shortened by Mario Carneiro, 19-Nov-2016) (Proof shortened by JJ, 7-Aug-2021)

Ref Expression
Hypotheses ralxfrd.1 ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐶 ) → 𝐴 ∈ 𝐵 )
ralxfrd.2 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ∃ 𝑦 ∈ 𝐶 𝑥 = 𝐴 )
ralxfrd.3 ⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
Assertion ralxfrd ( 𝜑 → ( ∀ 𝑥 ∈ 𝐵 𝜓 ↔ ∀ 𝑦 ∈ 𝐶 𝜒 ) )

Proof

Step Hyp Ref Expression
1 ralxfrd.1 ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐶 ) → 𝐴 ∈ 𝐵 )
2 ralxfrd.2 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ∃ 𝑦 ∈ 𝐶 𝑥 = 𝐴 )
3 ralxfrd.3 ⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
4 3 adantlr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐶 ) ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
5 1 4 rspcdv ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐶 ) → ( ∀ 𝑥 ∈ 𝐵 𝜓 → 𝜒 ) )
6 5 ralrimdva ⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐵 𝜓 → ∀ 𝑦 ∈ 𝐶 𝜒 ) )
7 r19.29 ⊢ ( ( ∀ 𝑦 ∈ 𝐶 𝜒 ∧ ∃ 𝑦 ∈ 𝐶 𝑥 = 𝐴 ) → ∃ 𝑦 ∈ 𝐶 ( 𝜒 ∧ 𝑥 = 𝐴 ) )
8 3 exbiri ⊢ ( 𝜑 → ( 𝑥 = 𝐴 → ( 𝜒 → 𝜓 ) ) )
9 8 impcomd ⊢ ( 𝜑 → ( ( 𝜒 ∧ 𝑥 = 𝐴 ) → 𝜓 ) )
10 9 rexlimdvw ⊢ ( 𝜑 → ( ∃ 𝑦 ∈ 𝐶 ( 𝜒 ∧ 𝑥 = 𝐴 ) → 𝜓 ) )
11 7 10 syl5 ⊢ ( 𝜑 → ( ( ∀ 𝑦 ∈ 𝐶 𝜒 ∧ ∃ 𝑦 ∈ 𝐶 𝑥 = 𝐴 ) → 𝜓 ) )
12 11 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( ( ∀ 𝑦 ∈ 𝐶 𝜒 ∧ ∃ 𝑦 ∈ 𝐶 𝑥 = 𝐴 ) → 𝜓 ) )
13 2 12 mpan2d ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( ∀ 𝑦 ∈ 𝐶 𝜒 → 𝜓 ) )
14 13 ralrimdva ⊢ ( 𝜑 → ( ∀ 𝑦 ∈ 𝐶 𝜒 → ∀ 𝑥 ∈ 𝐵 𝜓 ) )
15 6 14 impbid ⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐵 𝜓 ↔ ∀ 𝑦 ∈ 𝐶 𝜒 ) )