Theorem elabreximd 32605

Description: Class substitution in an image set. (Contributed by Thierry Arnoux, 30-Dec-2016.)

Hypotheses

Ref	Expression
elabreximd.1	⊢ Ⅎ𝑥𝜑
elabreximd.2	⊢ Ⅎ𝑥𝜒
elabreximd.3	⊢ (𝐴 = 𝐵 → (𝜒 ↔ 𝜓))
elabreximd.4	⊢ (𝜑 → 𝐴 ∈ 𝑉)
elabreximd.5	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝜓)

Assertion

Ref	Expression
elabreximd	⊢ ((𝜑 ∧ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑦 = 𝐵}) → 𝜒)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑦,𝐶

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)   𝐵(𝑥)   𝐶(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem elabreximd

Step	Hyp	Ref	Expression
1		elabreximd.4	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		eqeq1 2744	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑦 = 𝐵 ↔ 𝐴 = 𝐵))
3	2	rexbidv 3164	. . . . 5 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ 𝐶 𝑦 = 𝐵 ↔ ∃𝑥 ∈ 𝐶 𝐴 = 𝐵))
4	3	elabg 3621	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑦 = 𝐵} ↔ ∃𝑥 ∈ 𝐶 𝐴 = 𝐵))
5	1, 4	syl 17	. . 3 ⊢ (𝜑 → (𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑦 = 𝐵} ↔ ∃𝑥 ∈ 𝐶 𝐴 = 𝐵))
6	5	biimpa 477	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑦 = 𝐵}) → ∃𝑥 ∈ 𝐶 𝐴 = 𝐵)
7		elabreximd.1	. . . 4 ⊢ Ⅎ𝑥𝜑
8		elabreximd.2	. . . 4 ⊢ Ⅎ𝑥𝜒
9		simpr 485	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
10		elabreximd.5	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝜓)
11	10	adantr 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝐴 = 𝐵) → 𝜓)
12		elabreximd.3	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝜒 ↔ 𝜓))
13	12	biimpar 478	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝜓) → 𝜒)
14	9, 11, 13	syl2anc 590	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝐴 = 𝐵) → 𝜒)
15	14	exp31 420	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐶 → (𝐴 = 𝐵 → 𝜒)))
16	7, 8, 15	rexlimd 3247	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ 𝐶 𝐴 = 𝐵 → 𝜒))
17	16	imp 407	. 2 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐶 𝐴 = 𝐵) → 𝜒)
18	6, 17	syldan 597	1 ⊢ ((𝜑 ∧ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑦 = 𝐵}) → 𝜒)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547  Ⅎwnf 1790   ∈ wcel 2119  {cab 2718  ∃wrex 3064

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-12 2189  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065

This theorem is referenced by:  elabreximdv  32606  abrexss  32607  iinabrex  32665  disjabrex  32678  disjabrexf  32679