Theorem wl-equsald 38119

Description: Deduction version of equsal 2455. (Contributed by Wolf Lammen, 27-Jul-2019.)

Hypotheses

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

Assertion

Ref	Expression
wl-equsald	⊢ (𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜓) ↔ 𝜒))

Proof of Theorem wl-equsald

Step	Hyp	Ref	Expression
1		wl-equsald.2	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝜒)
2		19.23t 2252	. . 3 ⊢ (Ⅎ𝑥𝜒 → (∀𝑥(𝑥 = 𝑦 → 𝜒) ↔ (∃𝑥 𝑥 = 𝑦 → 𝜒)))
3	1, 2	syl 18	. 2 ⊢ (𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜒) ↔ (∃𝑥 𝑥 = 𝑦 → 𝜒)))
4		wl-equsald.1	. . 3 ⊢ Ⅎ𝑥𝜑
5		wl-equsald.3	. . . 4 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))
6	5	pm5.74d 276	. . 3 ⊢ (𝜑 → ((𝑥 = 𝑦 → 𝜓) ↔ (𝑥 = 𝑦 → 𝜒)))
7	4, 6	albid 2264	. 2 ⊢ (𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜓) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜒)))
8		ax6e 2421	. . . 4 ⊢ ∃𝑥 𝑥 = 𝑦
9	8	a1bi 365	. . 3 ⊢ (𝜒 ↔ (∃𝑥 𝑥 = 𝑦 → 𝜒))
10	9	a1i 11	. 2 ⊢ (𝜑 → (𝜒 ↔ (∃𝑥 𝑥 = 𝑦 → 𝜒)))
11	3, 7, 10	3bitr4d 314	1 ⊢ (𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜓) ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1565  ∃wex 1806  Ⅎwnf 1810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-12 2219  ax-13 2410

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-nf 1811

This theorem is referenced by:  wl-equsal  38121  wl-equsal1t  38122