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Theorem axc16i 2321
Description: Inference with axc16 2131 as its conclusion. (Contributed by NM, 20-May-2008.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
axc16i.1 (𝑥 = 𝑧 → (𝜑𝜓))
axc16i.2 (𝜓 → ∀𝑥𝜓)
Assertion
Ref Expression
axc16i (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
Distinct variable groups:   𝑥,𝑦,𝑧   𝜑,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦,𝑧)

Proof of Theorem axc16i
StepHypRef Expression
1 nfv 1840 . . 3 𝑧 𝑥 = 𝑦
2 nfv 1840 . . 3 𝑥 𝑧 = 𝑦
3 ax7 1940 . . 3 (𝑥 = 𝑧 → (𝑥 = 𝑦𝑧 = 𝑦))
41, 2, 3cbv3 2264 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑦)
5 ax7 1940 . . . . 5 (𝑧 = 𝑥 → (𝑧 = 𝑦𝑥 = 𝑦))
65spimv 2256 . . . 4 (∀𝑧 𝑧 = 𝑦𝑥 = 𝑦)
7 equcomi 1941 . . . . . 6 (𝑥 = 𝑦𝑦 = 𝑥)
8 equcomi 1941 . . . . . . 7 (𝑧 = 𝑦𝑦 = 𝑧)
9 ax7 1940 . . . . . . 7 (𝑦 = 𝑧 → (𝑦 = 𝑥𝑧 = 𝑥))
108, 9syl 17 . . . . . 6 (𝑧 = 𝑦 → (𝑦 = 𝑥𝑧 = 𝑥))
117, 10syl5com 31 . . . . 5 (𝑥 = 𝑦 → (𝑧 = 𝑦𝑧 = 𝑥))
1211alimdv 1842 . . . 4 (𝑥 = 𝑦 → (∀𝑧 𝑧 = 𝑦 → ∀𝑧 𝑧 = 𝑥))
136, 12mpcom 38 . . 3 (∀𝑧 𝑧 = 𝑦 → ∀𝑧 𝑧 = 𝑥)
14 equcomi 1941 . . . 4 (𝑧 = 𝑥𝑥 = 𝑧)
1514alimi 1736 . . 3 (∀𝑧 𝑧 = 𝑥 → ∀𝑧 𝑥 = 𝑧)
1613, 15syl 17 . 2 (∀𝑧 𝑧 = 𝑦 → ∀𝑧 𝑥 = 𝑧)
17 axc16i.1 . . . . 5 (𝑥 = 𝑧 → (𝜑𝜓))
1817biimpcd 239 . . . 4 (𝜑 → (𝑥 = 𝑧𝜓))
1918alimdv 1842 . . 3 (𝜑 → (∀𝑧 𝑥 = 𝑧 → ∀𝑧𝜓))
20 axc16i.2 . . . . 5 (𝜓 → ∀𝑥𝜓)
2120nf5i 2021 . . . 4 𝑥𝜓
22 nfv 1840 . . . 4 𝑧𝜑
2317biimprd 238 . . . . 5 (𝑥 = 𝑧 → (𝜓𝜑))
2414, 23syl 17 . . . 4 (𝑧 = 𝑥 → (𝜓𝜑))
2521, 22, 24cbv3 2264 . . 3 (∀𝑧𝜓 → ∀𝑥𝜑)
2619, 25syl6com 37 . 2 (∀𝑧 𝑥 = 𝑧 → (𝜑 → ∀𝑥𝜑))
274, 16, 263syl 18 1 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1478
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-10 2016  ax-11 2031  ax-12 2044  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-nf 1707
This theorem is referenced by:  axc16ALT  2365
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