Theorem elabreximdv 6351

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 1581	. 2 ⊢ Ⅎ𝑥𝜑
2		nfv 1581	. 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 6350	1 ⊢ ((𝜑 ∧ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑦 = 𝐵}) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  {cab 2224  ∃wrex 2529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823

This theorem is referenced by: (None)