Theorem elabreximd 28732

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 2614	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑦 = 𝐵 ↔ 𝐴 = 𝐵))
3	2	rexbidv 3034	. . . . 5 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ 𝐶 𝑦 = 𝐵 ↔ ∃𝑥 ∈ 𝐶 𝐴 = 𝐵))
4	3	elabg 3320	. . . 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 500	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑦 = 𝐵}) → ∃𝑥 ∈ 𝐶 𝐴 = 𝐵)
7		elabreximd.1	. . . 4 ⊢ Ⅎ𝑥𝜑
8		elabreximd.2	. . . 4 ⊢ Ⅎ𝑥𝜒
9		simpr 476	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
10		elabreximd.5	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝜓)
11	10	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝐴 = 𝐵) → 𝜓)
12		elabreximd.3	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝜒 ↔ 𝜓))
13	12	biimpar 501	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝜓) → 𝜒)
14	9, 11, 13	syl2anc 691	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝐴 = 𝐵) → 𝜒)
15	14	exp31 628	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐶 → (𝐴 = 𝐵 → 𝜒)))
16	7, 8, 15	rexlimd 3008	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ 𝐶 𝐴 = 𝐵 → 𝜒))
17	16	imp 444	. 2 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐶 𝐴 = 𝐵) → 𝜒)
18	6, 17	syldan 486	1 ⊢ ((𝜑 ∧ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑦 = 𝐵}) → 𝜒)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  {cab 2596  ∃wrex 2897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175

This theorem is referenced by:  elabreximdv  28733  abrexss  28734  disjabrex  28777  disjabrexf  28778