Theorem elabreximdv 33107

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 1946	. 2 ⊢ Ⅎ𝑥𝜑
2		nfv 1946	. 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 33106	1 ⊢ ((𝜑 ∧ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑦 = 𝐵}) → 𝜒)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {cab 2738  ∃wrex 3086

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087

This theorem is used by: (None)