Theorem cbvrmo 2834

Description: Change the bound variable of restricted "at most one" using implicit substitution. (Contributed by NM, 16-Jun-2017.)

Hypotheses

Ref	Expression
cbvral.1	⊢ Ⅎyφ
cbvral.2	⊢ Ⅎxψ
cbvral.3	⊢ (x = y → (φ ↔ ψ))

Assertion

Ref	Expression
cbvrmo	⊢ (∃*x ∈ A φ ↔ ∃*y ∈ A ψ)

Distinct variable groups:   x,A   y,A

Allowed substitution hints:   φ(x,y)   ψ(x,y)

Proof of Theorem cbvrmo

Step	Hyp	Ref	Expression
1		cbvral.1	. . . 4 ⊢ Ⅎyφ
2		cbvral.2	. . . 4 ⊢ Ⅎxψ
3		cbvral.3	. . . 4 ⊢ (x = y → (φ ↔ ψ))
4	1, 2, 3	cbvrex 2832	. . 3 ⊢ (∃x ∈ A φ ↔ ∃y ∈ A ψ)
5	1, 2, 3	cbvreu 2833	. . 3 ⊢ (∃!x ∈ A φ ↔ ∃!y ∈ A ψ)
6	4, 5	imbi12i 316	. 2 ⊢ ((∃x ∈ A φ → ∃!x ∈ A φ) ↔ (∃y ∈ A ψ → ∃!y ∈ A ψ))
7		rmo5 2827	. 2 ⊢ (∃*x ∈ A φ ↔ (∃x ∈ A φ → ∃!x ∈ A φ))
8		rmo5 2827	. 2 ⊢ (∃*y ∈ A ψ ↔ (∃y ∈ A ψ → ∃!y ∈ A ψ))
9	6, 7, 8	3bitr4i 268	1 ⊢ (∃*x ∈ A φ ↔ ∃*y ∈ A ψ)

Syntax hints:   → wi 4   ↔ wb 176  Ⅎwnf 1544  ∃wrex 2615  ∃!wreu 2616  ∃*wrmo 2617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622

This theorem is referenced by:  cbvrmov  2838