Theorem cbvrmo 2835

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

Hypotheses

Ref	Expression
cbvral.1	|- F/yph
cbvral.2	|- F/xps
cbvral.3	|- (x = y -> (ph <-> ps))

Assertion

Ref	Expression
cbvrmo	|- (E*x e. A ph <-> E*y e. A ps)

Distinct variable groups:   x,A   y,A

Allowed substitution hints:   ph(x,y)   ps(x,y)

Proof of Theorem cbvrmo

Step	Hyp	Ref	Expression
1		cbvral.1	. . . 4 |- F/yph
2		cbvral.2	. . . 4 |- F/xps
3		cbvral.3	. . . 4 |- (x = y -> (ph <-> ps))
4	1, 2, 3	cbvrex 2833	. . 3 |- (E.x e. A ph <-> E.y e. A ps)
5	1, 2, 3	cbvreu 2834	. . 3 |- (E!x e. A ph <-> E!y e. A ps)
6	4, 5	imbi12i 316	. 2 |- ((E.x e. A ph -> E!x e. A ph) <-> (E.y e. A ps -> E!y e. A ps))
7		rmo5 2828	. 2 |- (E*x e. A ph <-> (E.x e. A ph -> E!x e. A ph))
8		rmo5 2828	. 2 |- (E*y e. A ps <-> (E.y e. A ps -> E!y e. A ps))
9	6, 7, 8	3bitr4i 268	1 |- (E*x e. A ph <-> E*y e. A ps)

Syntax hints:   -> wi 4   <-> wb 176   F/wnf 1544  E.wrex 2616  E!wreu 2617  E*wrmo 2618

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 2479  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623

This theorem is referenced by:  cbvrmov  2839