Theorem wl-dral1d 35310

Description: A version of dral1 2438 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 35322 and nf5di 2288 show that this is in fact pointless. (Contributed by Wolf Lammen, 28-Jul-2019.)

Hypotheses

Ref	Expression
wl-dral1d.1	⊢ Ⅎ𝑥𝜑
wl-dral1d.2	⊢ Ⅎ𝑦𝜑
wl-dral1d.3	⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))

Assertion

Ref	Expression
wl-dral1d	⊢ (𝜑 → (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)))

Proof of Theorem wl-dral1d

Step	Hyp	Ref	Expression
1		wl-dral1d.3	. . . . . . . 8 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))
2	1	com12 32	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 → (𝜓 ↔ 𝜒)))
3	2	pm5.74d 276	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)))
4	3	sps 2185	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)))
5	4	dral1 2438	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝜑 → 𝜓) ↔ ∀𝑦(𝜑 → 𝜒)))
6		wl-dral1d.1	. . . . 5 ⊢ Ⅎ𝑥𝜑
7	6	19.21 2208	. . . 4 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))
8		wl-dral1d.2	. . . . 5 ⊢ Ⅎ𝑦𝜑
9	8	19.21 2208	. . . 4 ⊢ (∀𝑦(𝜑 → 𝜒) ↔ (𝜑 → ∀𝑦𝜒))
10	5, 7, 9	3bitr3g 316	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝜑 → ∀𝑥𝜓) ↔ (𝜑 → ∀𝑦𝜒)))
11	10	pm5.74rd 277	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)))
12	11	com12 32	1 ⊢ (𝜑 → (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1540  Ⅎwnf 1790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-10 2144  ax-12 2178  ax-13 2371

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-ex 1787  df-nf 1791

This theorem is referenced by:  wl-cbvalnaed  35311