Theorem rmoeq1f 3431

Description: Equality theorem for restricted at-most-one quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Hypotheses

Ref	Expression
rmoeq1f.1	⊢ Ⅎ𝑥𝐴
rmoeq1f.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
rmoeq1f	⊢ (𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐵 𝜑))

Proof of Theorem rmoeq1f

Step	Hyp	Ref	Expression
1		rmoeq1f.1	. . . 4 ⊢ Ⅎ𝑥𝐴
2		rmoeq1f.2	. . . 4 ⊢ Ⅎ𝑥𝐵
3	1, 2	nfeq 2922	. . 3 ⊢ Ⅎ𝑥 𝐴 = 𝐵
4		eleq2 2833	. . . 4 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
5	4	anbi1d 630	. . 3 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ 𝜑)))
6	3, 5	mobid 2553	. 2 ⊢ (𝐴 = 𝐵 → (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)))
7		df-rmo 3388	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
8		df-rmo 3388	. 2 ⊢ (∃*𝑥 ∈ 𝐵 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
9	6, 7, 8	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  Ⅎwnfc 2893  ∃*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-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-mo 2543  df-cleq 2732  df-clel 2819  df-nfc 2895  df-rmo 3388

This theorem is referenced by:  reueq1f  3432