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Theorem wl-equsal1i 34860
Description: The antecedent 𝑥 = 𝑦 is irrelevant, if one or both setvar variables are not free in 𝜑. (Contributed by Wolf Lammen, 1-Sep-2018.)
Hypotheses
Ref Expression
wl-equsal1i.1 (Ⅎ𝑥𝜑 ∨ Ⅎ𝑦𝜑)
wl-equsal1i.2 (𝑥 = 𝑦𝜑)
Assertion
Ref Expression
wl-equsal1i 𝜑

Proof of Theorem wl-equsal1i
StepHypRef Expression
1 wl-equsal1i.1 . 2 (Ⅎ𝑥𝜑 ∨ Ⅎ𝑦𝜑)
2 wl-equsal1i.2 . . 3 (𝑥 = 𝑦𝜑)
32gen2 1798 . 2 𝑥𝑦(𝑥 = 𝑦𝜑)
4 sp 2184 . . . . 5 (∀𝑦𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
54alcoms 2163 . . . 4 (∀𝑥𝑦(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
6 wl-equsal1t 34858 . . . 4 (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜑))
75, 6syl5ib 247 . . 3 (Ⅎ𝑥𝜑 → (∀𝑥𝑦(𝑥 = 𝑦𝜑) → 𝜑))
8 wl-equsalcom 34859 . . . . 5 (∀𝑦(𝑦 = 𝑥𝜑) ↔ ∀𝑦(𝑥 = 𝑦𝜑))
9 wl-equsal1t 34858 . . . . . 6 (Ⅎ𝑦𝜑 → (∀𝑦(𝑦 = 𝑥𝜑) ↔ 𝜑))
109biimpd 232 . . . . 5 (Ⅎ𝑦𝜑 → (∀𝑦(𝑦 = 𝑥𝜑) → 𝜑))
118, 10syl5bir 246 . . . 4 (Ⅎ𝑦𝜑 → (∀𝑦(𝑥 = 𝑦𝜑) → 𝜑))
1211spsd 2188 . . 3 (Ⅎ𝑦𝜑 → (∀𝑥𝑦(𝑥 = 𝑦𝜑) → 𝜑))
137, 12jaoi 854 . 2 ((Ⅎ𝑥𝜑 ∨ Ⅎ𝑦𝜑) → (∀𝑥𝑦(𝑥 = 𝑦𝜑) → 𝜑))
141, 3, 13mp2 9 1 𝜑
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 844  wal 1536  wnf 1785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-10 2146  ax-11 2162  ax-12 2179  ax-13 2392
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786
This theorem is referenced by: (None)
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