Theorem elabreximdv 32970

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 1947	. 2 ⊢ Ⅎ𝑥𝜑
2		nfv 1947	. 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 32969	1 ⊢ ((𝜑 ∧ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑦 = 𝐵}) → 𝜒)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2740  ∃wrex 3088

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-12 2215  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089

This theorem is used by: (None)