Theorem ralab2 2903

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

Hypothesis

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

Assertion

Ref	Expression
ralab2	⊢ (∀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 ralab2

Step	Hyp	Ref	Expression
1		df-ral 2460	. 2 ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜓 ↔ ∀𝑥(𝑥 ∈ {𝑦 ∣ 𝜑} → 𝜓))
2		nfsab1 2167	. . . 4 ⊢ Ⅎ𝑦 𝑥 ∈ {𝑦 ∣ 𝜑}
3		nfv 1528	. . . 4 ⊢ Ⅎ𝑦𝜓
4	2, 3	nfim 1572	. . 3 ⊢ Ⅎ𝑦(𝑥 ∈ {𝑦 ∣ 𝜑} → 𝜓)
5		nfv 1528	. . 3 ⊢ Ⅎ𝑥(𝜑 → 𝜒)
6		eleq1 2240	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ 𝑦 ∈ {𝑦 ∣ 𝜑}))
7		abid 2165	. . . . 5 ⊢ (𝑦 ∈ {𝑦 ∣ 𝜑} ↔ 𝜑)
8	6, 7	bitrdi 196	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ 𝜑))
9		ralab2.1	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))
10	8, 9	imbi12d 234	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ {𝑦 ∣ 𝜑} → 𝜓) ↔ (𝜑 → 𝜒)))
11	4, 5, 10	cbval 1754	. 2 ⊢ (∀𝑥(𝑥 ∈ {𝑦 ∣ 𝜑} → 𝜓) ↔ ∀𝑦(𝜑 → 𝜒))
12	1, 11	bitri 184	1 ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜓 ↔ ∀𝑦(𝜑 → 𝜒))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1351   ∈ wcel 2148  {cab 2163  ∀wral 2455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-ral 2460

This theorem is referenced by:  ralrab2  2904  ssintab  3863