Metamath Proof Explorer


Theorem rmoeq1OLD

Description: Obsolete version of rmoeq1 as of 12-Mar-2025. (Contributed by Alexander van der Vekens, 17-Jun-2017) Remove usage of ax-10 , ax-11 , and ax-12 . (Revised by Steven Nguyen, 30-Apr-2023) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion rmoeq1OLD ( 𝐴 = 𝐵 → ( ∃* 𝑥𝐴 𝜑 ↔ ∃* 𝑥𝐵 𝜑 ) )

Proof

Step Hyp Ref Expression
1 eleq2 ( 𝐴 = 𝐵 → ( 𝑥𝐴𝑥𝐵 ) )
2 1 anbi1d ( 𝐴 = 𝐵 → ( ( 𝑥𝐴𝜑 ) ↔ ( 𝑥𝐵𝜑 ) ) )
3 2 mobidv ( 𝐴 = 𝐵 → ( ∃* 𝑥 ( 𝑥𝐴𝜑 ) ↔ ∃* 𝑥 ( 𝑥𝐵𝜑 ) ) )
4 df-rmo ( ∃* 𝑥𝐴 𝜑 ↔ ∃* 𝑥 ( 𝑥𝐴𝜑 ) )
5 df-rmo ( ∃* 𝑥𝐵 𝜑 ↔ ∃* 𝑥 ( 𝑥𝐵𝜑 ) )
6 3 4 5 3bitr4g ( 𝐴 = 𝐵 → ( ∃* 𝑥𝐴 𝜑 ↔ ∃* 𝑥𝐵 𝜑 ) )