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Theorem ralab2 3512
Description: Universal quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
Hypothesis
Ref Expression
ralab2.1 (𝑥 = 𝑦 → (𝜓𝜒))
Assertion
Ref Expression
ralab2 (∀𝑥 ∈ {𝑦𝜑}𝜓 ↔ ∀𝑦(𝜑𝜒))
Distinct variable groups:   𝑥,𝑦   𝜒,𝑥   𝜑,𝑥   𝜓,𝑦
Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑥)   𝜒(𝑦)

Proof of Theorem ralab2
StepHypRef Expression
1 df-ral 3055 . 2 (∀𝑥 ∈ {𝑦𝜑}𝜓 ↔ ∀𝑥(𝑥 ∈ {𝑦𝜑} → 𝜓))
2 nfsab1 2750 . . . 4 𝑦 𝑥 ∈ {𝑦𝜑}
3 nfv 1992 . . . 4 𝑦𝜓
42, 3nfim 1974 . . 3 𝑦(𝑥 ∈ {𝑦𝜑} → 𝜓)
5 nfv 1992 . . 3 𝑥(𝜑𝜒)
6 eleq1w 2822 . . . . 5 (𝑥 = 𝑦 → (𝑥 ∈ {𝑦𝜑} ↔ 𝑦 ∈ {𝑦𝜑}))
7 abid 2748 . . . . 5 (𝑦 ∈ {𝑦𝜑} ↔ 𝜑)
86, 7syl6bb 276 . . . 4 (𝑥 = 𝑦 → (𝑥 ∈ {𝑦𝜑} ↔ 𝜑))
9 ralab2.1 . . . 4 (𝑥 = 𝑦 → (𝜓𝜒))
108, 9imbi12d 333 . . 3 (𝑥 = 𝑦 → ((𝑥 ∈ {𝑦𝜑} → 𝜓) ↔ (𝜑𝜒)))
114, 5, 10cbval 2416 . 2 (∀𝑥(𝑥 ∈ {𝑦𝜑} → 𝜓) ↔ ∀𝑦(𝜑𝜒))
121, 11bitri 264 1 (∀𝑥 ∈ {𝑦𝜑}𝜓 ↔ ∀𝑦(𝜑𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1630  wcel 2139  {cab 2746  wral 3050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-clel 2756  df-ral 3055
This theorem is referenced by:  ralrab2  3513  ssintab  4646  efgval  18330  efger  18331  elintabg  38382  elintima  38447
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