Theorem rexxfr3dALT 35607

Description: Longer proof of rexxfr3d 35606 using ax-11 2158 instead of ax-12 2178, without the disjoint variable condition 𝐴𝑥𝑦. (Contributed by SN, 19-Jun-2025.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
rexxfr3dALT.s	⊢ (𝑥 = 𝑋 → (𝜓 ↔ 𝜒))
rexxfr3dALT.x	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ ∃𝑦 ∈ 𝐵 𝑥 = 𝑋))
rexxfr3dALT.a	⊢ (𝜑 → 𝑋 ∈ 𝑉)

Assertion

Ref	Expression
rexxfr3dALT	⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒))

Distinct variable groups:   𝜑,𝑥,𝑦   𝜓,𝑦   𝜒,𝑥   𝑥,𝑋   𝑥,𝐵

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑦)   𝑉(𝑥,𝑦)   𝑋(𝑦)

Proof of Theorem rexxfr3dALT

Step	Hyp	Ref	Expression
1		rexxfr3dALT.x	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ ∃𝑦 ∈ 𝐵 𝑥 = 𝑋))
2	1	anbi1d 630	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (∃𝑦 ∈ 𝐵 𝑥 = 𝑋 ∧ 𝜓)))
3		rexxfr3dALT.s	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝜓 ↔ 𝜒))
4	3	pm5.32i 574	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝜓) ↔ (𝑥 = 𝑋 ∧ 𝜒))
5	4	rexbii 3100	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 (𝑥 = 𝑋 ∧ 𝜓) ↔ ∃𝑦 ∈ 𝐵 (𝑥 = 𝑋 ∧ 𝜒))
6		r19.41v 3195	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 (𝑥 = 𝑋 ∧ 𝜓) ↔ (∃𝑦 ∈ 𝐵 𝑥 = 𝑋 ∧ 𝜓))
7	5, 6	bitr3i 277	. . . . 5 ⊢ (∃𝑦 ∈ 𝐵 (𝑥 = 𝑋 ∧ 𝜒) ↔ (∃𝑦 ∈ 𝐵 𝑥 = 𝑋 ∧ 𝜓))
8	2, 7	bitr4di 289	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃𝑦 ∈ 𝐵 (𝑥 = 𝑋 ∧ 𝜒)))
9	8	exbidv 1920	. . 3 ⊢ (𝜑 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃𝑥∃𝑦 ∈ 𝐵 (𝑥 = 𝑋 ∧ 𝜒)))
10		df-rex 3077	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
11		19.41v 1949	. . . . 5 ⊢ (∃𝑥(𝑥 = 𝑋 ∧ 𝜒) ↔ (∃𝑥 𝑥 = 𝑋 ∧ 𝜒))
12	11	rexbii 3100	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑥(𝑥 = 𝑋 ∧ 𝜒) ↔ ∃𝑦 ∈ 𝐵 (∃𝑥 𝑥 = 𝑋 ∧ 𝜒))
13		rexcom4 3294	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑥(𝑥 = 𝑋 ∧ 𝜒) ↔ ∃𝑥∃𝑦 ∈ 𝐵 (𝑥 = 𝑋 ∧ 𝜒))
14	12, 13	bitr3i 277	. . 3 ⊢ (∃𝑦 ∈ 𝐵 (∃𝑥 𝑥 = 𝑋 ∧ 𝜒) ↔ ∃𝑥∃𝑦 ∈ 𝐵 (𝑥 = 𝑋 ∧ 𝜒))
15	9, 10, 14	3bitr4g 314	. 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐵 (∃𝑥 𝑥 = 𝑋 ∧ 𝜒)))
16		rexxfr3dALT.a	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
17		elisset 2826	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → ∃𝑥 𝑥 = 𝑋)
18	16, 17	syl 17	. . . 4 ⊢ (𝜑 → ∃𝑥 𝑥 = 𝑋)
19	18	biantrurd 532	. . 3 ⊢ (𝜑 → (𝜒 ↔ (∃𝑥 𝑥 = 𝑋 ∧ 𝜒)))
20	19	rexbidv 3185	. 2 ⊢ (𝜑 → (∃𝑦 ∈ 𝐵 𝜒 ↔ ∃𝑦 ∈ 𝐵 (∃𝑥 𝑥 = 𝑋 ∧ 𝜒)))
21	15, 20	bitr4d 282	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108  ∃wrex 3076

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-11 2158

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-clel 2819  df-rex 3077

This theorem is referenced by: (None)