Theorem rmoeq1OLD 3427

Description: Obsolete version of rmoeq1 3425 as of 12-Mar-2025. (Contributed by Alexander van der Vekens, 17-Jun-2017.) Remove usage of ax-10 2141, ax-11 2158, and ax-12 2178. (Revised by Steven Nguyen, 30-Apr-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
rmoeq1OLD	⊢ (𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐵 𝜑))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rmoeq1OLD

Step	Hyp	Ref	Expression
1		eleq2 2833	. . . 4 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
2	1	anbi1d 630	. . 3 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ 𝜑)))
3	2	mobidv 2552	. 2 ⊢ (𝐴 = 𝐵 → (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)))
4		df-rmo 3388	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
5		df-rmo 3388	. 2 ⊢ (∃*𝑥 ∈ 𝐵 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
6	3, 4, 5	3bitr4g 314	1 ⊢ (𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐵 𝜑))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∃*wmo 2541  ∃*wrmo 3387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-mo 2543  df-cleq 2732  df-clel 2819  df-rmo 3388

This theorem is referenced by: (None)