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Theorem wl-dral1d 34639
Description: A version of dral1 2458 with a context. Note: At first glance one might be tempted to generalize this (or a similar) theorem by weakening the first two hypotheses adding a 𝑥 = 𝑦, 𝑥𝑥 = 𝑦 or 𝜑 antecedent. wl-equsal1i 34651 and nf5di 2287 show that this is in fact pointless. (Contributed by Wolf Lammen, 28-Jul-2019.)
Hypotheses
Ref Expression
wl-dral1d.1 𝑥𝜑
wl-dral1d.2 𝑦𝜑
wl-dral1d.3 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
Assertion
Ref Expression
wl-dral1d (𝜑 → (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)))

Proof of Theorem wl-dral1d
StepHypRef Expression
1 wl-dral1d.3 . . . . . . . 8 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
21com12 32 . . . . . . 7 (𝑥 = 𝑦 → (𝜑 → (𝜓𝜒)))
32pm5.74d 274 . . . . . 6 (𝑥 = 𝑦 → ((𝜑𝜓) ↔ (𝜑𝜒)))
43sps 2176 . . . . 5 (∀𝑥 𝑥 = 𝑦 → ((𝜑𝜓) ↔ (𝜑𝜒)))
54dral1 2458 . . . 4 (∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝜑𝜓) ↔ ∀𝑦(𝜑𝜒)))
6 wl-dral1d.1 . . . . 5 𝑥𝜑
7619.21 2200 . . . 4 (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))
8 wl-dral1d.2 . . . . 5 𝑦𝜑
9819.21 2200 . . . 4 (∀𝑦(𝜑𝜒) ↔ (𝜑 → ∀𝑦𝜒))
105, 7, 93bitr3g 314 . . 3 (∀𝑥 𝑥 = 𝑦 → ((𝜑 → ∀𝑥𝜓) ↔ (𝜑 → ∀𝑦𝜒)))
1110pm5.74rd 275 . 2 (∀𝑥 𝑥 = 𝑦 → (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)))
1211com12 32 1 (𝜑 → (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wal 1528  wnf 1777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-10 2138  ax-12 2169  ax-13 2385
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-ex 1774  df-nf 1778
This theorem is referenced by:  wl-cbvalnaed  34640
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