Theorem wl-dral1d

Description: A version of dral1 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 , A. x x = y or ph antecedent. wl-equsal1i and nf5di show that this is in fact pointless. (Contributed by Wolf Lammen, 28-Jul-2019)

	Ref	Expression
Hypotheses	wl-dral1d.1	⊢ Ⅎ 𝑥 𝜑
	wl-dral1d.2	⊢ Ⅎ 𝑦 𝜑
	wl-dral1d.3	⊢ ( 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓 ↔ 𝜒 ) ) )
Assertion	wl-dral1d	⊢ ( 𝜑 → ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝜓 ↔ ∀ 𝑦 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		wl-dral1d.1	⊢ Ⅎ 𝑥 𝜑
2		wl-dral1d.2	⊢ Ⅎ 𝑦 𝜑
3		wl-dral1d.3	⊢ ( 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓 ↔ 𝜒 ) ) )
4	3	com12	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) )
5	4	pm5.74d	⊢ ( 𝑥 = 𝑦 → ( ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → 𝜒 ) ) )
6	5	sps	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → 𝜒 ) ) )
7	6	dral1	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ↔ ∀ 𝑦 ( 𝜑 → 𝜒 ) ) )
8	1	19.21	⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∀ 𝑥 𝜓 ) )
9	2	19.21	⊢ ( ∀ 𝑦 ( 𝜑 → 𝜒 ) ↔ ( 𝜑 → ∀ 𝑦 𝜒 ) )
10	7 8 9	3bitr3g	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ( 𝜑 → ∀ 𝑥 𝜓 ) ↔ ( 𝜑 → ∀ 𝑦 𝜒 ) ) )
11	10	pm5.74rd	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝜑 → ( ∀ 𝑥 𝜓 ↔ ∀ 𝑦 𝜒 ) ) )
12	11	com12	⊢ ( 𝜑 → ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝜓 ↔ ∀ 𝑦 𝜒 ) ) )