Theorem rmoeq1 3118

Description: Equality theorem for restricted uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Assertion

Ref	Expression
rmoeq1	⊢ (𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐵 𝜑))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rmoeq1

Step	Hyp	Ref	Expression
1		nfcv 2751	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2751	. 2 ⊢ Ⅎ𝑥𝐵
3	1, 2	rmoeq1f 3114	1 ⊢ (𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐵 𝜑))

Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475  ∃*wrmo 2899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-eu 2462  df-mo 2463  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rmo 2904

This theorem is referenced by:  rmoeqd  3126  poimirlem2  32581