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Theorem cbv3hv 1850
 Description: Lemma for ax10 1944. Similar to cbv3h 1983. Requires distinct variables but avoids ax-12 1925. (Contributed by NM, 25-Jul-2015.) (Proof shortened by Wolf Lammen, 29-Dec-2017.)
Hypotheses
Ref Expression
cbv3hv.1 (φyφ)
cbv3hv.2 (ψxψ)
cbv3hv.3 (x = y → (φψ))
Assertion
Ref Expression
cbv3hv (xφyψ)
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)   ψ(x,y)

Proof of Theorem cbv3hv
StepHypRef Expression
1 cbv3hv.1 . . 3 (φyφ)
21alimi 1559 . 2 (xφxyφ)
3 a9ev 1656 . . . . . . 7 x x = y
4 cbv3hv.3 . . . . . . . 8 (x = y → (φψ))
54eximi 1576 . . . . . . 7 (x x = yx(φψ))
63, 5ax-mp 5 . . . . . 6 x(φψ)
7619.35i 1601 . . . . 5 (xφxψ)
8 cbv3hv.2 . . . . . 6 (ψxψ)
9819.9h 1780 . . . . 5 (xψψ)
107, 9sylib 188 . . . 4 (xφψ)
1110alimi 1559 . . 3 (yxφyψ)
1211a7s 1735 . 2 (xyφyψ)
132, 12syl 15 1 (xφyψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1540  ∃wex 1541 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545 This theorem is referenced by: (None)
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