Theorem ralab2OLD 3685

Description: Obsolete version of ralab2 3684 as of 1-Dec-2023. (Contributed by Mario Carneiro, 3-Sep-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

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

Assertion

Ref	Expression
ralab2OLD	⊢ (∀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 ralab2OLD

Step	Hyp	Ref	Expression
1		df-ral 3142	. 2 ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜓 ↔ ∀𝑥(𝑥 ∈ {𝑦 ∣ 𝜑} → 𝜓))
2		nfsab1 2807	. . . 4 ⊢ Ⅎ𝑦 𝑥 ∈ {𝑦 ∣ 𝜑}
3		nfv 1914	. . . 4 ⊢ Ⅎ𝑦𝜓
4	2, 3	nfim 1896	. . 3 ⊢ Ⅎ𝑦(𝑥 ∈ {𝑦 ∣ 𝜑} → 𝜓)
5		nfv 1914	. . 3 ⊢ Ⅎ𝑥(𝜑 → 𝜒)
6		eleq1w 2894	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ 𝑦 ∈ {𝑦 ∣ 𝜑}))
7		abid 2802	. . . . 5 ⊢ (𝑦 ∈ {𝑦 ∣ 𝜑} ↔ 𝜑)
8	6, 7	syl6bb 289	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ 𝜑))
9		ralab2.1	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))
10	8, 9	imbi12d 347	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ {𝑦 ∣ 𝜑} → 𝜓) ↔ (𝜑 → 𝜒)))
11	4, 5, 10	cbvalv1 2360	. 2 ⊢ (∀𝑥(𝑥 ∈ {𝑦 ∣ 𝜑} → 𝜓) ↔ ∀𝑦(𝜑 → 𝜒))
12	1, 11	bitri 277	1 ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜓 ↔ ∀𝑦(𝜑 → 𝜒))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1534   ∈ wcel 2113  {cab 2798  ∀wral 3137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-10 2144  ax-11 2160  ax-12 2176

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2799  df-clel 2892  df-ral 3142

This theorem is referenced by: (None)