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Metamath Proof Explorer

Theorem reuxfr1dd

Description: Transfer existential uniqueness from a variable x to another variable y contained in expression A . Simplifies reuxfr1d . (Contributed by Zhi Wang, 20-Sep-2025)

Ref Expression
Hypotheses reuxfr1dd.1 ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐶 ) → 𝐴 ∈ 𝐵 )
reuxfr1dd.2 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ∃! 𝑦 ∈ 𝐶 𝑥 = 𝐴 )
reuxfr1dd.3 ⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴 ) ) → ( 𝜓 ↔ 𝜒 ) )
Assertion reuxfr1dd ( 𝜑 → ( ∃! 𝑥 ∈ 𝐵 𝜓 ↔ ∃! 𝑦 ∈ 𝐶 𝜒 ) )

Proof

Step Hyp Ref Expression
1 reuxfr1dd.1 ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐶 ) → 𝐴 ∈ 𝐵 )
2 reuxfr1dd.2 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ∃! 𝑦 ∈ 𝐶 𝑥 = 𝐴 )
3 reuxfr1dd.3 ⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴 ) ) → ( 𝜓 ↔ 𝜒 ) )
4 reurex ⊢ ( ∃! 𝑦 ∈ 𝐶 𝑥 = 𝐴 → ∃ 𝑦 ∈ 𝐶 𝑥 = 𝐴 )
5 2 4 syl ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ∃ 𝑦 ∈ 𝐶 𝑥 = 𝐴 )
6 5 biantrurd ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( 𝜓 ↔ ( ∃ 𝑦 ∈ 𝐶 𝑥 = 𝐴 ∧ 𝜓 ) ) )
7 r19.41v ⊢ ( ∃ 𝑦 ∈ 𝐶 ( 𝑥 = 𝐴 ∧ 𝜓 ) ↔ ( ∃ 𝑦 ∈ 𝐶 𝑥 = 𝐴 ∧ 𝜓 ) )
8 3 pm5.32da ⊢ ( 𝜑 → ( ( ( 𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴 ) ∧ 𝜓 ) ↔ ( ( 𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴 ) ∧ 𝜒 ) ) )
9 anass ⊢ ( ( ( 𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴 ) ∧ 𝜓 ) ↔ ( 𝑦 ∈ 𝐶 ∧ ( 𝑥 = 𝐴 ∧ 𝜓 ) ) )
10 anass ⊢ ( ( ( 𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴 ) ∧ 𝜒 ) ↔ ( 𝑦 ∈ 𝐶 ∧ ( 𝑥 = 𝐴 ∧ 𝜒 ) ) )
11 8 9 10 3bitr3g ⊢ ( 𝜑 → ( ( 𝑦 ∈ 𝐶 ∧ ( 𝑥 = 𝐴 ∧ 𝜓 ) ) ↔ ( 𝑦 ∈ 𝐶 ∧ ( 𝑥 = 𝐴 ∧ 𝜒 ) ) ) )
12 11 rexbidv2 ⊢ ( 𝜑 → ( ∃ 𝑦 ∈ 𝐶 ( 𝑥 = 𝐴 ∧ 𝜓 ) ↔ ∃ 𝑦 ∈ 𝐶 ( 𝑥 = 𝐴 ∧ 𝜒 ) ) )
13 7 12 bitr3id ⊢ ( 𝜑 → ( ( ∃ 𝑦 ∈ 𝐶 𝑥 = 𝐴 ∧ 𝜓 ) ↔ ∃ 𝑦 ∈ 𝐶 ( 𝑥 = 𝐴 ∧ 𝜒 ) ) )
14 13 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( ( ∃ 𝑦 ∈ 𝐶 𝑥 = 𝐴 ∧ 𝜓 ) ↔ ∃ 𝑦 ∈ 𝐶 ( 𝑥 = 𝐴 ∧ 𝜒 ) ) )
15 6 14 bitrd ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( 𝜓 ↔ ∃ 𝑦 ∈ 𝐶 ( 𝑥 = 𝐴 ∧ 𝜒 ) ) )
16 15 reubidva ⊢ ( 𝜑 → ( ∃! 𝑥 ∈ 𝐵 𝜓 ↔ ∃! 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 ( 𝑥 = 𝐴 ∧ 𝜒 ) ) )
17 reurmo ⊢ ( ∃! 𝑦 ∈ 𝐶 𝑥 = 𝐴 → ∃* 𝑦 ∈ 𝐶 𝑥 = 𝐴 )
18 2 17 syl ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ∃* 𝑦 ∈ 𝐶 𝑥 = 𝐴 )
19 1 18 reuxfrd ⊢ ( 𝜑 → ( ∃! 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 ( 𝑥 = 𝐴 ∧ 𝜒 ) ↔ ∃! 𝑦 ∈ 𝐶 𝜒 ) )
20 16 19 bitrd ⊢ ( 𝜑 → ( ∃! 𝑥 ∈ 𝐵 𝜓 ↔ ∃! 𝑦 ∈ 𝐶 𝜒 ) )