Theorem rexabOLD 3685

Description: Obsolete version of rexab 3684 as of 2-Nov-2024. (Contributed by Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro, 3-Sep-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
ralab.1	⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓))

Assertion

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

Distinct variable groups:   𝑥,𝑦   𝜓,𝑦

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

Proof of Theorem rexabOLD

Step	Hyp	Ref	Expression
1		df-rex 3060	. 2 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∃𝑥(𝑥 ∈ {𝑦 ∣ 𝜑} ∧ 𝜒))
2		vex 3468	. . . . 5 ⊢ 𝑥 ∈ V
3		ralab.1	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓))
4	2, 3	elab 3663	. . . 4 ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ 𝜓)
5	4	anbi1i 624	. . 3 ⊢ ((𝑥 ∈ {𝑦 ∣ 𝜑} ∧ 𝜒) ↔ (𝜓 ∧ 𝜒))
6	5	exbii 1847	. 2 ⊢ (∃𝑥(𝑥 ∈ {𝑦 ∣ 𝜑} ∧ 𝜒) ↔ ∃𝑥(𝜓 ∧ 𝜒))
7	1, 6	bitri 275	1 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∃𝑥(𝜓 ∧ 𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∃wex 1778   ∈ wcel 2107  {cab 2712  ∃wrex 3059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-rex 3060  df-v 3466

This theorem is referenced by: (None)