Theorem rexab2 3628

Description: Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)

Hypothesis

Ref	Expression
ralab2.1	⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
rexab2	⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜓 ↔ ∃𝑦(𝜑 ∧ 𝜒))

Distinct variable groups:   𝑥,𝑦   𝜒,𝑥   𝜑,𝑥   𝜓,𝑦

Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑥)   𝜒(𝑦)

Proof of Theorem rexab2

Step	Hyp	Ref	Expression
1		df-rex 3111	. 2 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜓 ↔ ∃𝑥(𝑥 ∈ {𝑦 ∣ 𝜑} ∧ 𝜓))
2		nfsab1 2784	. . . 4 ⊢ Ⅎ𝑦 𝑥 ∈ {𝑦 ∣ 𝜑}
3		nfv 1892	. . . 4 ⊢ Ⅎ𝑦𝜓
4	2, 3	nfan 1881	. . 3 ⊢ Ⅎ𝑦(𝑥 ∈ {𝑦 ∣ 𝜑} ∧ 𝜓)
5		nfv 1892	. . 3 ⊢ Ⅎ𝑥(𝜑 ∧ 𝜒)
6		eleq1w 2865	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ 𝑦 ∈ {𝑦 ∣ 𝜑}))
7		abid 2779	. . . . 5 ⊢ (𝑦 ∈ {𝑦 ∣ 𝜑} ↔ 𝜑)
8	6, 7	syl6bb 288	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ 𝜑))
9		ralab2.1	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))
10	8, 9	anbi12d 630	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ {𝑦 ∣ 𝜑} ∧ 𝜓) ↔ (𝜑 ∧ 𝜒)))
11	4, 5, 10	cbvexv1 2321	. 2 ⊢ (∃𝑥(𝑥 ∈ {𝑦 ∣ 𝜑} ∧ 𝜓) ↔ ∃𝑦(𝜑 ∧ 𝜒))
12	1, 11	bitri 276	1 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜓 ↔ ∃𝑦(𝜑 ∧ 𝜒))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∃wex 1761   ∈ wcel 2081  {cab 2775  ∃wrex 3106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-10 2112  ax-11 2126  ax-12 2141

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-clel 2863  df-rex 3111

This theorem is referenced by:  rexrab2  3629  tmdgsum2  22388  clrellem  39467  brtrclfv2  39557