Theorem hmphsymv 10537

Description: A more general version of hmphsym 10529. Certainly not a useful proof (since it's a simple consequence of hmpher 10536) but it shows that the conditions of hmphsym 10529 could (and should) be weakened.

Hypotheses

Ref	Expression
hmphsymv.1	|- A e. V
hmphsymv.2	|- B e. V

Assertion

Ref	Expression
hmphsymv	|- (A ~= B <-> B ~= A)

Proof of Theorem hmphsymv

Step	Hyp	Ref	Expression
1		hmphsymv.1	. 2 |- A e. V
2		hmphsymv.2	. 2 |- B e. V
3		hmpher 10536	. 2 |- Er ~=
4	1, 2, 3	ersymb 4273	1 |- (A ~= B <-> B ~= A)

Syntax hints:   <-> wb 146   e. wcel 958  Vcvv 1811   class class class wbr 2619   ~= chomeo 10514

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-opr 3965  df-oprab 3966  df-er 4261  df-homeo 10515  df-hmph 10523

Copyright terms: Public domain