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Theorem ax16ALT 1859
Description: Version of ax16 1813 that does not require ax-10 1505 or ax12 1512 for its proof. (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax16ALT (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem ax16ALT
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 sbequ12 1771 . 2 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
2 ax-17 1526 . . 3 (𝜑 → ∀𝑧𝜑)
32hbsb3 1808 . 2 ([𝑧 / 𝑥]𝜑 → ∀𝑥[𝑧 / 𝑥]𝜑)
41, 3ax16i 1858 1 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1351  [wsb 1762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763
This theorem is referenced by:  dvelimALT  2010  dvelimfv  2011
  Copyright terms: Public domain W3C validator