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Theorem wl-equsald 32954
Description: Deduction version of equsal 2290. (Contributed by Wolf Lammen, 27-Jul-2019.)
Hypotheses
Ref Expression
wl-equsald.1 𝑥𝜑
wl-equsald.2 (𝜑 → Ⅎ𝑥𝜒)
wl-equsald.3 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
Assertion
Ref Expression
wl-equsald (𝜑 → (∀𝑥(𝑥 = 𝑦𝜓) ↔ 𝜒))

Proof of Theorem wl-equsald
StepHypRef Expression
1 wl-equsald.2 . . 3 (𝜑 → Ⅎ𝑥𝜒)
2 19.23t 2077 . . 3 (Ⅎ𝑥𝜒 → (∀𝑥(𝑥 = 𝑦𝜒) ↔ (∃𝑥 𝑥 = 𝑦𝜒)))
31, 2syl 17 . 2 (𝜑 → (∀𝑥(𝑥 = 𝑦𝜒) ↔ (∃𝑥 𝑥 = 𝑦𝜒)))
4 wl-equsald.1 . . 3 𝑥𝜑
5 wl-equsald.3 . . . 4 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
65pm5.74d 262 . . 3 (𝜑 → ((𝑥 = 𝑦𝜓) ↔ (𝑥 = 𝑦𝜒)))
74, 6albid 2088 . 2 (𝜑 → (∀𝑥(𝑥 = 𝑦𝜓) ↔ ∀𝑥(𝑥 = 𝑦𝜒)))
8 ax6e 2249 . . . 4 𝑥 𝑥 = 𝑦
98a1bi 352 . . 3 (𝜒 ↔ (∃𝑥 𝑥 = 𝑦𝜒))
109a1i 11 . 2 (𝜑 → (𝜒 ↔ (∃𝑥 𝑥 = 𝑦𝜒)))
113, 7, 103bitr4d 300 1 (𝜑 → (∀𝑥(𝑥 = 𝑦𝜓) ↔ 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1478  wex 1701  wnf 1705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-12 2044  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1702  df-nf 1707
This theorem is referenced by:  wl-equsal  32955  wl-equsal1t  32956  wl-sb6rft  32959
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