Theorem rmoeq1f 3381

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 2914	. . 3 ⊢ Ⅎ𝑥 𝐴 = 𝐵
4		eleq2 2828	. . . 4 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
5	4	anbi1d 637	. . 3 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ 𝜑)))
6	3, 5	mobid 2554	. 2 ⊢ (𝐴 = 𝐵 → (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)))
7		df-rmo 3344	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
8		df-rmo 3344	. 2 ⊢ (∃*𝑥 ∈ 𝐵 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
9	6, 7, 8	3bitr4g 315	1 ⊢ (𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐵 𝜑))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∃*wmo 2541  Ⅎwnfc 2886  ∃*wrmo 3343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-nf 1791  df-mo 2543  df-cleq 2731  df-clel 2814  df-nfc 2888  df-rmo 3344

This theorem is referenced by:  reueq1f  3382