Theorem elabreximdv 30273

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

Hypotheses

Ref	Expression
elabreximdv.1	⊢ (𝐴 = 𝐵 → (𝜒 ↔ 𝜓))
elabreximdv.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)
elabreximdv.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝜓)

Assertion

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

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

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

Proof of Theorem elabreximdv

Step	Hyp	Ref	Expression
1		nfv 1915	. 2 ⊢ Ⅎ𝑥𝜑
2		nfv 1915	. 2 ⊢ Ⅎ𝑥𝜒
3		elabreximdv.1	. 2 ⊢ (𝐴 = 𝐵 → (𝜒 ↔ 𝜓))
4		elabreximdv.2	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
5		elabreximdv.3	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝜓)
6	1, 2, 3, 4, 5	elabreximd 30272	1 ⊢ ((𝜑 ∧ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑦 = 𝐵}) → 𝜒)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  {cab 2801  ∃wrex 3141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146

This theorem is referenced by: (None)