Theorem moxfr 42650

Description: Transfer at-most-one between related expressions. (Contributed by Stefan O'Rear, 12-Feb-2015.)

Hypotheses

Ref	Expression
moxfr.a	⊢ 𝐴 ∈ V
moxfr.b	⊢ ∃!𝑦 𝑥 = 𝐴
moxfr.c	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
moxfr	⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)

Distinct variable groups:   𝜓,𝑥   𝜑,𝑦   𝑥,𝐴   𝑥,𝑦

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

Proof of Theorem moxfr

Step	Hyp	Ref	Expression
1		moxfr.a	. . . . . 6 ⊢ 𝐴 ∈ V
2	1	a1i 11	. . . . 5 ⊢ (𝑦 ∈ V → 𝐴 ∈ V)
3		moxfr.b	. . . . . . . 8 ⊢ ∃!𝑦 𝑥 = 𝐴
4		euex 2580	. . . . . . . 8 ⊢ (∃!𝑦 𝑥 = 𝐴 → ∃𝑦 𝑥 = 𝐴)
5	3, 4	ax-mp 5	. . . . . . 7 ⊢ ∃𝑦 𝑥 = 𝐴
6		rexv 3517	. . . . . . 7 ⊢ (∃𝑦 ∈ V 𝑥 = 𝐴 ↔ ∃𝑦 𝑥 = 𝐴)
7	5, 6	mpbir 231	. . . . . 6 ⊢ ∃𝑦 ∈ V 𝑥 = 𝐴
8	7	a1i 11	. . . . 5 ⊢ (𝑥 ∈ V → ∃𝑦 ∈ V 𝑥 = 𝐴)
9		moxfr.c	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
10	2, 8, 9	rexxfr 5434	. . . 4 ⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑦 ∈ V 𝜓)
11		rexv 3517	. . . 4 ⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑)
12		rexv 3517	. . . 4 ⊢ (∃𝑦 ∈ V 𝜓 ↔ ∃𝑦𝜓)
13	10, 11, 12	3bitr3i 301	. . 3 ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)
14	1, 3, 9	euxfrw 3743	. . 3 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)
15	13, 14	imbi12i 350	. 2 ⊢ ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (∃𝑦𝜓 → ∃!𝑦𝜓))
16		moeu 2586	. 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
17		moeu 2586	. 2 ⊢ (∃*𝑦𝜓 ↔ (∃𝑦𝜓 → ∃!𝑦𝜓))
18	15, 16, 17	3bitr4i 303	1 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537  ∃wex 1777   ∈ wcel 2108  ∃*wmo 2541  ∃!weu 2571  ∃wrex 3076  Vcvv 3488

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-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-v 3490

This theorem is referenced by: (None)